13.2E: Exercises for Limits and Continuity

13.2E: Exercises for Limits and Continuity

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13.2: Limits and Continuity

1) Use the limit laws for functions of two variables to evaluate each limit below, given that (displaystyle lim_{(x,y)→(a,b)}f(x,y) = 5) and (displaystyle lim_{(x,y)→(a,b)}g(x,y) = 2).

  1. (displaystyle lim_{(x,y)→(a,b)}left[f(x,y) + g(x,y) ight])
  2. (displaystyle lim_{(x,y)→(a,b)}left[f(x,y) g(x,y) ight])
  3. (displaystyle lim_{(x,y)→(a,b)}left[ dfrac{7f(x,y)}{g(x,y)} ight])
  4. (displaystyle lim_{(x,y)→(a,b)}left[dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)} ight])
  1. (displaystyle lim_{(x,y)→(a,b)}left[f(x,y) + g(x,y) ight] = displaystyle lim_{(x,y)→(a,b)}f(x,y) + displaystyle lim_{(x,y)→(a,b)}g(x,y)= 5 + 2 = 7)
  2. (displaystyle lim_{(x,y)→(a,b)}left[f(x,y) g(x,y) ight] =left(displaystyle lim_{(x,y)→(a,b)}f(x,y) ight) left(displaystyle lim_{(x,y)→(a,b)}g(x,y) ight) = 5(2) = 10)
  3. (displaystyle lim_{(x,y)→(a,b)}left[ dfrac{7f(x,y)}{g(x,y)} ight] = frac{7left(displaystyle lim_{(x,y)→(a,b)}f(x,y) ight)}{displaystyle lim_{(x,y)→(a,b)}g(x,y)}=frac{7(5)}{2} = 17.5)
  4. (displaystyle lim_{(x,y)→(a,b)}left[dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)} ight] = frac{2left(displaystyle lim_{(x,y)→(a,b)}f(x,y) ight) - 4 left(displaystyle lim_{(x,y)→(a,b)}g(x,y) ight)}{displaystyle lim_{(x,y)→(a,b)}f(x,y) - displaystyle lim_{(x,y)→(a,b)}g(x,y)}= frac{2(5) - 4(2)}{5 - 2} = frac{2}{3})

In exercises 2 - 4, find the limit of the function.

2) (displaystyle lim_{(x,y)→(1,2)}x)

3) (displaystyle lim_{(x,y)→(1,2)}frac{5x^2y}{x^2+y^2})

(displaystyle lim_{(x,y)→(1,2)}frac{5x^2y}{x^2+y^2} = 2)

4) Show that the limit (displaystyle lim_{(x,y)→(0,0)}frac{5x^2y}{x^2+y^2}) exists and is the same along the paths: (y)-axis and (x)-axis, and along ( y=x).

In exercises 5 - 19, evaluate the limits at the indicated values of (x) and (y). If the limit does not exist, state this and explain why the limit does not exist.

5) (displaystyle lim_{(x,y)→(0,0)}frac{4x^2+10y^2+4}{4x^2−10y^2+6})

(displaystyle lim_{(x,y)→(0,0)}frac{4x^2+10y^2+4}{4x^2−10y^2+6} = frac{2}{3} )

6) (displaystyle lim_{(x,y)→(11,13)}sqrt{frac{1}{xy}})

7) (displaystyle lim_{(x,y)→(0,1)}frac{y^2sin x}{x})

(displaystyle lim_{(x,y)→(0,1)}frac{y^2sin x}{x} = 1)

8) (displaystyle lim_{(x,y)→(0,0)}sin(frac{x^8+y^7}{x−y+10}))

9) (displaystyle lim_{(x,y)→(π/4,1)}frac{y an x}{y+1})

(displaystyle lim_{(x,y)→(π/4,1)}frac{y an x}{y+1}=frac{1}{2})

10) (displaystyle lim_{(x,y)→(0,π/4)}frac{sec x+2}{3x− an y})

11) (displaystyle lim_{(x,y)→(2,5)}(frac{1}{x}−frac{5}{y}))

(displaystyle lim_{(x,y)→(2,5)}(frac{1}{x}−frac{5}{y}) = −frac{1}{2})

12) (displaystyle lim_{(x,y)→(4,4)}xln y)

13) (displaystyle lim_{(x,y)→(4,4)}e^{−x^2−y^2})

(displaystyle lim_{(x,y)→(4,4)}e^{−x^2−y^2} = e^{−32})

14) (displaystyle lim_{(x,y)→(0,0)}sqrt{9−x^2−y^2})

15) (displaystyle lim_{(x,y)→(1,2)}(x^2y^3−x^3y^2+3x+2y))

(displaystyle lim_{(x,y)→(1,2)}(x^2y^3−x^3y^2+3x+2y) = 11)

16) (displaystyle lim_{(x,y)→(π,π)}xsin(frac{x+y}{4}))

17) (displaystyle lim_{(x,y)→(0,0)}frac{xy+1}{x^2+y^2+1})

(displaystyle lim_{(x,y)→(0,0)}frac{xy+1}{x^2+y^2+1} = 1)

18) (displaystyle lim_{(x,y)→(0,0)}frac{x^2+y^2}{sqrt{x^2+y^2+1}−1})

19) (displaystyle lim_{(x,y)→(0,0)}ln(x^2+y^2))

The limit does not exist because when (x) and (y) both approach zero, the function approaches ( ln 0), which is undefined (approaches negative infinity).

In exercises 20 - 21, complete the statement.

20) A point ( (x_0,y_0)) in a plane region ( R) is an interior point of (R) if _________________.

21) A point ( (x_0,y_0)) in a plane region (R) is called a boundary point of (R) if ___________.

Every open disk centered at ( (x_0,y_0)) contains points inside ( R) and outside ( R).

In exercises 22 - 25, use algebraic techniques to evaluate the limit.

22) (displaystyle lim_{(x,y)→(2,1)}frac{x−y−1}{sqrt{x−y}−1})

23) (displaystyle lim_{(x,y)→(0,0)}frac{x^4−4y^4}{x^2+2y^2})

(displaystyle lim_{(x,y)→(0,0)}frac{x^4−4y^4}{x^2+2y^2} = 0)

24) (displaystyle lim_{(x,y)→(0,0)}frac{x^3−y^3}{x−y})

25) (displaystyle lim_{(x,y)→(0,0)}frac{x^2−xy}{sqrt{x}−sqrt{y}})

(displaystyle lim_{(x,y)→(0,0)}frac{x^2−xy}{sqrt{x}−sqrt{y}} = 0)

In exercises 26 - 27, evaluate the limits of the functions of three variables.

26) (displaystyle lim_{(x,y,z)→(1,2,3)}frac{xz^2−y^2z}{xyz−1})

27) (displaystyle lim_{(x,y,z)→(0,0,0)}frac{x^2−y^2−z^2}{x^2+y^2−z^2})

The limit does not exist.

In exercises 28 - 31, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not.

28) (displaystyle lim_{(x,y)→(0,0)}frac{xy+y^3}{x^2+y^2})

a. Along the (x)-axis ( (y=0))

b. Along the (y)-axis ( (x=0))

c. Along the path (y=2x)

29) Evaluate (displaystyle lim_{(x,y)→(0,0)}frac{xy+y^3}{x^2+y^2}) using the results of previous problem.

The limit does not exist. The function approaches two different values along different paths.

30) (displaystyle lim_{(x,y)→(0,0)}frac{x^2y}{x^4+y^2})

a. Along the path (y=x^2)

31) Evaluate (displaystyle lim_{(x,y)→(0,0)}frac{x^2y}{x^4+y^2}) using the results of previous problem.

The limit does not exist because the function approaches two different values along the paths.

In exercises 32 - 35, discuss the continuity of each function. Find the largest region in the (xy)-plane in which each function is continuous.

32) ( f(x,y)=sin(xy))

33) ( f(x,y)=ln(x+y))

The function ( f) is continuous in the region ( y>−x.)

34) ( f(x,y)=e^{3xy})

35) ( f(x,y)=dfrac{1}{xy})

The function (f) is continuous at all points in the (xy)-plane except at points on the (x)- and (y)-axes.

In exercises 36 - 38, determine the region in which the function is continuous. Explain your answer.

36) ( f(x,y)=dfrac{x^2y}{x^2+y^2})

37) ( f(x,y)=)( egin{cases}dfrac{x^2y}{x^2+y^2} & if(x,y)≠(0,0) & if(x,y)=(0,0)end{cases})

Show that the function approaches different values along two different paths.
The function is continuous at ( (0,0)) since the limit of the function at ( (0,0)) is ( 0), the same value of ( f(0,0).)

38) ( f(x,y)=dfrac{sin(x^2+y^2)}{x^2+y^2})

39) Determine whether ( g(x,y)=dfrac{x^2−y^2}{x^2+y^2}) is continuous at ( (0,0)).

The function is discontinuous at ( (0,0).) The limit at ( (0,0)) fails to exist and ( g(0,0)) does not exist.

40) Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which ( f(x,y)=dfrac{1}{x^2−y}) is continuous.

41) Determine the region of the (xy)-plane in which the composite function ( g(x,y)=arctan(frac{xy^2}{x+y})) is continuous. Use technology to support your conclusion.

Since the function ( arctan x) is continuous over ( (−∞,∞), g(x,y)=arctan(frac{xy^2}{x+y})) is continuous where ( z=dfrac{xy^2}{x+y}) is continuous. The inner function ( z) is continuous on all points of the (xy)-plane except where ( y=−x.) Thus, ( g(x,y)=arctan(frac{xy^2}{x+y})) is continuous on all points of the coordinate plane except at points at which ( y=−x.)

42) Determine the region of the (xy)-plane in which ( f(x,y)=ln(x^2+y^2−1)) is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for ( x) and ( y) carefully!)

43) At what points in space is ( g(x,y,z)=x^2+y^2−2z^2) continuous?

All points ( P(x,y,z)) in space

44) At what points in space is ( g(x,y,z)=dfrac{1}{x^2+z^2−1}) continuous?

45) Show that (displaystyle lim_{(x,y)→(0,0)}frac{1}{x^2+y^2}) does not exist at ( (0,0)) by plotting the graph of the function.


The graph increases without bound as ( x) and ( y) both approach zero.

46) [T] Evaluate (displaystyle lim_{(x,y)→(0,0)}frac{−xy^2}{x^2+y^4}) by plotting the function using a CAS. Determine analytically the limit along the path ( x=y^2.)

47) [T]

a. Use a CAS to draw a contour map of ( z=sqrt{9−x^2−y^2}).

b. What is the name of the geometric shape of the level curves?

c. Give the general equation of the level curves.

d. What is the maximum value of ( z)?

e. What is the domain of the function?

f. What is the range of the function?



b. The level curves are circles centered at ( (0,0)) with radius ( 9−c).
c. ( x^2+y^2=9−c)
d. ( z=3)
e. ( {(x,y)∈R^2∣x^2+y^2≤9})
f. ( {z|0≤z≤3})

48) True or False: If we evaluate (displaystyle lim_{(x,y)→(0,0)}f(x)) along several paths and each time the limit is ( 1), we can conclude that (displaystyle lim_{(x,y)→(0,0)}f(x)=1.)

49) Use polar coordinates to find (displaystyle lim_{(x,y)→(0,0)}frac{sinsqrt{x^2+y^2}}{sqrt{x^2+y^2}}.) You can also find the limit using L’Hôpital’s rule.

(displaystyle lim_{(x,y)→(0,0)}frac{sinsqrt{x^2+y^2}}{sqrt{x^2+y^2}} = 1)

50) Use polar coordinates to find (displaystyle lim_{(x,y)→(0,0)}cos(x^2+y^2).)

51) Discuss the continuity of ( f(g(x,y))) where ( f(t)=1/t) and ( g(x,y)=2x−5y.)

( f(g(x,y))) is continuous at all points ( (x,y)) that are not on the line ( 2x−5y=0.)

52) Given ( f(x,y)=x^2−4y,) find (displaystyle lim_{h→0}frac{f(x+h,y)−f(x,y)}{h}.)

53) Given ( f(x,y)=x^2−4y,) find (displaystyle lim_{h→0}frac{f(1+h,y)−f(1,y)}{h}).

( displaystyle lim_{h→0}frac{f(1+h,y)−f(1,y)}{h} = 2)


  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at

  • Paul Seeburger (Monroe Community College) edited the LaTeX and created problem 1.

Change management exercises are games or simulations that motivate employees to accept change and actively engage with the change process. When posed in a positive, zero-stakes setting, these exercises minimize resistance and make change a fun event.

The exercises below are just a few examples of activities that will get employees excited about upcoming change.

1. Cross Your Arms the “Other” Way

Gather employees and ask them to cross their arms. Then, once they are comfortable in that position, ask them to fold their arms the other way. It feels quite different, doesn’t it? Despite the fact that they are only making a slight change to their stance, the feeling is not the same.

Ask your employees to discuss how this small but noticeable change makes them feel. For some, the unfamiliarity of the posture might be frustrating. Or, it might just be a bit uncomfortable. However, the longer they sit with their arms this way, the more comfortable they’ll become. Relate this feeling to how an organizational change can feel wrong at first, but begins to feel more natural as time goes on.

2. The Alien at Dinner

In this game, employees pretend that they are an alien sitting at a human dinner party for the first time. While observing the humans around them, the alien employee is asked to note the odd behavior that the humans exhibit while they eat and talk to one another.

This exercise demonstrates to employees the importance of diversity in thought, keeping an open mind, and really considering others’ ideas. It helps people learn to question what they have long accepted as normal. Change management exercises like this help staff become more comfortable with assessing how things are done now and how they can be improved.

3. Changing Places

Arrange chairs in a circle and place an object in the center. Ask employees to take a seat, then observe the object. After a minute or so, ask them to get up and change seats. Call on them to describe the object from their new point of view. Then, tell them they are allowed to get up and change seats once more.

Some employees will wish to stay put. However, staying in the same place limits the number of perspectives that they can have. In contrast, each time the employees observe the object from a different perspective, they have the opportunity to notice something new. Change management exercises that illustrate the importance of gaining a new perspective help mollify resistance and show how a change can be beneficial.

4. The Ups and Downs of Change

Create a list of change-related words, such as “transformation,” “implementation,” “transition,” “training,” “process change,” and more of the like. Read different words aloud and ask employees to step forward if the word evokes a positive response, backwards for a negative response.

After each word, have employees observe the changes in the room and discuss why they chose to step forward or backward. Those who stepped backward may have a stronger tendency to resist change, or at least associate negative emotions with change. Open a dialogue about how thinking of change-related terms in a positive way will actually help them get them farther.

5. The Four P’s

With a large sheet of paper or board, create four columns labeled with each of the following words: Project, Purpose, Particulars, and People.

Ask the group to fill out each column depending on how they believe a given change will affect those four entities. By asking employees to articulate their concerns about how a change will affect certain things, change managers can more effectively address their apprehensions. Discussing these concerns will help employees gain a better understanding of the true effects of the change and reduce resistance.

6. Bounce Back

Ask your employees to pair up. Then give each pair a rubber ball and ask them to bounce it back and forth. After a few minutes, ask the group if they were ever concerned that the ball would not bounce up after they tossed it to the ground.

Just like a bouncy ball, organizations will rebound from the challenges produced from change. Change management exercises such as this encourage employees to embrace the movement and understanding that they will recover afterwards, even if a change is uncomfortable at the moment.

7. Can-Do Company

Split the group of employees into teams of 5 or 6 and ask them to come up with a simple, fun business idea that they will present to the entire group. Give each team member a role like planning, design or sales.

After letting the groups strategize for 10 minutes, move a few participants from each group around to other groups. Then introduce one new criterium the business idea must contain. Allow the groups to strategize for another 10 minutes, given the new information. At the end of the session, each group presents their idea and everyone votes on the best one.

This exercise demonstrates the importance of being flexible during the planning process. It simulates the need to work as a team, even amid changes to the team itself, and embracing others’ ideas. After, ask the employees to reflect on what good things came out of having a new perspective on the team. How did the end product change from that first round of planning to the second?

Class 12 students often face pressure when board exams are right on the corner Moreover, as the exam gets closer, it is not the subject matter that you require, but it is the final revision notes that prove to be a great help. Revision is not just referring to the notes you made in class but certainly, revision notes should include all the important topics of the chapter.

Be it theory or numerical questions, Continuity and Differentiability notes provided by Vedantu would help you in attempting the questions based on continuity and differentiation with ease.

It is often seen the students don’t prepare the revision notes while going through the chapter. And, while revising the chapter they tend to miss out various important topics. Due to this, we at Vedantu initiate to provide Continuity and Differentiability Class 12 notes which include some practical guidelines to make your preparation easier.

Hence, it is advised to refer to Continuity and Differentiability notes as it includes all the important concepts and formulas that will help you to solve all the numerical questions given in Class 12 Chapter 5 easily.

13.2E: Exercises for Limits and Continuity

In this lesson you will explore continuity at a point, investigate discontinuity at a point, display discontinuities, and learn how to redefine a function to remove a point discontinuity. You will then use the TI-83 to graph piecewise defined functions.

Informally, a function is said to be continuous on an interval if you can sketch its graph on the interval without lifting your pencil off the paper. The formal definition of continuity starts by defining continuity at a point and then extends to continuity on an interval. The formal definition may not seem to have much in common with the concept of sketching a graph without lifting your pencil off the paper, but after investigating several examples with your TI-83, the connection between the formal and informal definitions should be more apparent.

Continuity at a Point and on an Interval

The formal definition of continuity at a point has three conditions that must be met.

A function f(x) is continuous at a point where x = c if

A function is continuous on an interval if it is continuous at every point in the interval.

Discontinuity at a Point

The definition for continuity at a point may make more sense as you see it applied to functions with discontinuities. If any of the three conditions in the definition of continuity fails when x = c, the function is discontinuous at that point. Examine the continuity of when x = 0.

Checking the Conditions for Continuity

By the definition of continuity, you can conclude that is not continuous at x = 0.

Displaying Discontinuities

A discontinuity at a point may be illustrated by graphing the function in an appropriate window. The discontinuity only shows up if it is at an x-value used in the plot. It is difficult (it might be impossible) to force it to show up at a point like or .

The y-axis will need to be turned off in order to see the discontinuity at x = 0.

The discontinuity is represented as a hole in the graph at the point with coordinates (0,1).

Although the discontinuity appears as a hole in the graph, it could be argued that no hole should appear because the missing point is infinitely small. On the TI-83 the missing point is represented by a missing pixel only if the x-value of the hole is an x-value used in the plot.

Removing the Discontinuity

The following shows how can be redefined to create a new function that is exactly like the original function for all non-zero values of x, but is continuous at x = 0.

Define a new function g(x) to be the function whose values are for and y = 1 for x = 0.

This new function is called a piecewise function because different formulas are applied to different parts of the domain. The graph of g(x) is the same as the graph of except it includes the point (0,1), the point that fills the hole.

Graphing a Piecewise Function

You can graph the piecewise function by entering the two pieces in Y1 and Y2. The first piece should already be in Y1. ( Y1 = sin(X)/X )

The denominator of Y2 = 1/(X=0) is a Boolean expression because it is either true or false. When the Boolean expression is true, it returns a value of 1. When it is false, the Boolean expression returns a value of 0. This means that Y2 will equal 1 when x = 0 and it will be undefined when because division by zero is not defined.

The hole at (0, 1) has been filled.

Turning the Axes Back On

Before leaving this lesson you should turn the graphing axes on.

8.1.1 Redefine to make it continuous at x = 2. Click here for the answer.

13.2E: Exercises for Limits and Continuity

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The COOP Planning Guide contains hyperlinks to facilitate navigation within the document and to external sources. Each of the individual sections of the Guide is outlined below.
Section I: COOP Planning for State Courts

COOP stands for continuity of operations courts develop a COOP plan to ensure they know what to do if faced with an emergency that threatens continuation of normal operations. Traditionally, a COOP plan is developed and implemented for situations in which the courthouse or court-related facilities are threatened or inaccessible (e.g., as a result of a natural or man-made disaster). A traditional COOP plan establishes effective processes and procedures to quickly deploy pre-designated personnel, equipment, vital records and supporting hardware and software to an alternative site to sustain organizational operations for up to 30 days. It also covers the resumption of normal operations after the emergency has ended.

Section II: COOP Planning Steps

The steps listed below help a court develop a COOP capability in the event of a manmade or natural disaster. Each step includes an explanation of what needs to be done and links to additional resources, if necessary

Step 1: Initiate the planning process

Step 1a: Provide leadership and develop infrastructure
Step 1b: Review court&rsquos legal authority in COOP planning and execution
Step 1c: Gather information on related COOP planning activities
Step 1d: Specify planning assumptions
Step 1e: Consider potential disaster scenarios

Step 2: Prepare COOP plan elements
Step 2a: Identify & prioritize essential functions
Step 2b: Determine essential functions staff
Step 2c: Establish orders of succession and delegate authorities
Step 2d: Identify alternate facilities
Step 2e: Identify business practices to limit personal contact
Step 2f: Identify communications methods
Step 2g: Ensure interoperable communications
Step 2h: Identify vital records & databases
Step 2i: Develop resources to manage human capital
Step 2j: Prepare drive-away kits
Step 2k: Plan devolution process

Step 3: Prepare COOP plan procedures

Step 3a: Phase I procedures for COOP plan activation
Step 3b: Phase I procedures for alert and notification
Step 3c: Phase I procedures for transition to an alternate facility
Step 3d: Phase II procedures for alternate facility operations
Step 3e: Phase III procedures for reconstitution
Step 3f: Modified procedures for a pandemic

Step 4: Complete the plan template

Step 5: Maintain and practice plan

Section III: COOP Plan Worksheets

The following Worksheets are provided to help courts gather information critical to preparing their COOP plan. The Worksheets are linked to various steps discussed under Section II COOP Planning Steps. Worksheets are available for editing by clicking the Microsoft Word icon above.

Worksheet A: Determine Essential Functions

Worksheet B: Priority of Essential Functions

Worksheet C: Essential Functions Staff

Worksheet D: COOP Staff Roster

Worksheet E: Orders of Succession and Delegation of Authorities

Worksheet F: Contact Information for Key Decision-makers and Successors

Worksheet G: Alternate Work Site Requirements

Worksheet H: Alternate Work Site Options

Worksheet I: Alternate Work Sites by Disaster Scenarios

Worksheet J: Potential Strategies to Limit Personal Contact

Worksheet K: Strategies to Limit Personal Contact for Each Essential Function

Worksheet L: Communications Plan

Worksheet M: Media Contacts

Worksheet N: Interoperability of Communications Systems

Worksheet O: Inventory of Vital Records

Worksheet P: Restoration Resources

Worksheet Q: Staff Directory

Worksheet R: Emergency Contacts for Staff

Worksheet S: Emergency Services Available

Worksheet T: Personnel Policies

Worksheet U: Drive-Away Kits

Worksheet V: Devolution Plan

Worksheet W: COOP Plan Testing Program

Worksheet X: COOP Plan Training Program

Worksheet Y: COOP Plan Exercise Program

Section IV: COOP Plan Template

This template offers courts a guide for preparing their own continuity of operations (COOP) plan. Each section describes the information that should be included and, in some cases, offers language that can be adapted to fit individual courts. Suggested language is italicized information to be added by the court is bracketed.

Many of the sections require courts to work through several steps and make decisions before the section can be completed. These sections are linked to information and worksheets provided in Continuity of Court Operations: Steps for COOP Planning to help courts in this process.

Outreach and Technical Assistance

In coordination with the FEMA Regions, FEMA’s National Continuity Programs provides outreach and technical assistance to whole community partners across the nation. Outreach and technical assistance offerings is available to non-federal partners, including state, local, tribal and territorial governments the private sector non-governmental organizations and critical infrastructure owners and operators.

Continuity Assessment Tool

Jurisdictions are encouraged to complete an assessment of current plans and programs using the Continuity Assessment Tool to identify shortfalls or gaps to guide requests for technical assistance.

Continuity Resource Toolkit and Guidance Circular

FEMA maintains the Continuity Resource Toolkit, which is designed to provide additional tools, templates, and resources to assist in implementing the concepts found within the Continuity Guidance Circular.

If interested in outreach or technical assistance opportunities, please contact [email protected]

Additional Resources

Learn about the Continuity Excellence Series, two levels of training courses designed to address the full spectrum of requirements to support a viable continuity capability.

Oblique asymptotes

Oblique asymptotes are only calculated if there are no horizontal asymptotes.

Like the other two types of asymptotes, oblique asymptotes are oblique straight lines, to which the function gets closer and closer, but never touches.

As it is an oblique line, it has this shape:

And it is about calculating the coefficients m and n to find the equation of the line.

To calculate the coefficient m we use the following formula:

For the oblique asymptote to exist, m cannot be equal to zero, since if m=0, the asymptote would be horizontal:

The coefficient m cannot be infinite either, because otherwise the asymptote would be vertical:

The coefficient n is calculated with the following formula:

And finally, once the values of the coefficients m and n have been obtained, we would have the equation of the line that defines the oblique asymptote:

Use Continuity to evaluate the limit

Now, i know that lim sin(x + sin(x)) = 0. However, i don't understand what it means to use Continuity to evaluate this limit. What does it want me to say? Should i go through the "3 step" for continuity? If i do the 3-step, is it assuming that i'm looking to see if the function is continuous at pi? I am simply confused as to what it is asking me to do

Also, i know that this question was answered before, a few years ago. It still leaves me confused on what to do, so i would appreciate any help


Elite Member

If the problem says use continuity, then, no, you are being asked to prove that sin(x) is continuous. It is saying that you may assume that you have already proved that sin(x) is continuous for all x.

Now, do you know what "continuous" means?? The simplest way or putting it, without a lot of technical points, is that a function, f(x), is continuous at x= a if and only if (displaystyle lim_ f(x)= f(a)). Because, of course, (displaystyle a= lim_ x), that can also be written in the very useful form (displaystyle lim_f(x)= f(lim_ x)).

Here, that means that (displaystyle lim_ sin(x+ sin(x))= sin(lim_(x+ sin(x))). Can you complete that?

13.2E: Exercises for Limits and Continuity

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4. &thinspSee Trader Update dated January 29, 2015, available here:​pillar.

5. &thinspNYSE Arca Equities is a wholly-owned corporation of NYSE Arca and operates as a facility of NYSE Arca.

6. &thinspIn connection with the NYSE Arca implementation of Pillar, NYSE Arca filed four rule proposals relating to Pillar. See Securities Exchange Act Release Nos. 74951 (May 13, 2015), 80 FR 28721 (May 19, 2015) (Notice) and 75494 (July 20, 2015), 80 FR 44170 (July 24, 2015) (SR-NYSEArca-2015-38) (Approval Order of NYSE Arca Pillar I Filing, adopting rules for Trading Sessions, Order Ranking and Display, and Order Execution) Securities Exchange Act Release Nos. 75497 (July 21, 2015), 80 FR 45022 (July 28, 2015) (Notice) and 76267 (October 26, 2015), 80 FR 66951 (October 30, 2015) (SR-NYSEArca-2015-56) (Approval Order of NYSE Arca Pillar II Filing, adopting rules for Orders and Modifiers and the Retail Liquidity Program) Securities Exchange Act Release Nos. 75467 (July 16, 2015), 80 FR 43515 (July 22, 2015) (Notice) and 76198 (October 20, 2015), 80 FR 65274 (October 26, 2015) (SR-NYSEArca-2015-58) (Approval Order of NYSE Arca Pillar III Filing, adopting rules for Trading Halts, Short Sales, Limit Up-Limit Down, and Odd Lots and Mixed Lots) and Securities Exchange Act Release Nos. 76085 (October 6, 2015), 80 FR 61513 (October 13, 2015) (Notice) and 76869 (January 11, 2016), 81 FR 2276 (January 15, 2016) (Approval Order of NYSE Arca Pillar IV Filing, adopting rules for Auctions).

8. &thinspSee Securities Exchange Act Release No. 79242 (November 4, 2016), 81 FR 79081 (November 10, 2016) (SR-NYSEMKT-2016-97) (Notice and Filing of Immediate Effectiveness of Proposed Rule Change) (the &ldquoFramework Filing&rdquo).

9. &thinspTo distinguish Rule 1E-13E from Exchange rules that govern options trading, the Exchange proposes a non-substantive change to amend the description of &ldquoPillar Platform Rules&rdquo after Rule 0&mdashEquities to specify that these are &ldquocash equities&rdquo rules.

10. &thinspSee Securities Exchange Act Release No. 79400 (November 25, 2016), 81 FR 86750 (December 1, 2016) (SR-NYSEMKT-2016-103) (Notice) (the &ldquoETP Listing Rules Filing&rdquo). When trading on Pillar, the Exchange would not be relying on Rule 500&mdashEquities&mdashRule 525&mdashEquities for authority to trade securities on an unlisted trading privileges basis. Accordingly, the Exchange proposes to amend Rule 500&mdashEquities to provide that the Rules of that series (Rules 500&mdashEquities&mdashRule 525&mdashEquities) would not be applicable to trading on the Pillar trading platform. To use terms applicable to trading on Pillar, the Exchange also proposes to amend Rule 2A(b)(2)&mdashEquities to replace the term &ldquoNasdaq Security&rdquo with the term &ldquoUTP Security&rdquo and replace the rule reference from Rule 501&mdashEquities to Rule 1.1E(ii).

11. &thinspRules 1E-13E are including in the &ldquoEquities Rules&rdquo portion of the Exchange's rule book. Pursuant to Rule 0&mdashEquities, the Equities Rules govern all transactions conducted on the Equities Trading Systems.

12. &thinspThe Exchange proposes to amend the description of Cash Equities Pillar Platform Rules, which precedes Rule 1E, to delete the last sentence, which currently provides that &ldquo[t]he following rules will not be applicable to trading on the Pillar trading platform: Rules 7&mdashEquities, 55&mdashEquities, 56&mdashEquities, 62&mdashEquities, and 80B&mdashEquities.&rdquo As proposed, the inapplicability of these rules on the Pillar platform would be addressed in the preamble that the Exchange proposes to add to each of these rules. The Exchange further proposes to retain Rule 56&mdashEquities when the Exchange migrates to Pillar, as it addresses the unit of trading for rights, which are listed on the Exchange.

13. &thinspBecause these non-substantive differences would be applied throughout the proposed rules, the Exchange will not note these differences separately for each proposed rule.

14. &thinspRule 123C(1)(e)&mdashEquities sets forth how the Exchange currently determines the Official Closing Price of a security listed on the Exchange.

15. &thinspThe Exchange will file a separate proposed rule change to specify fees for cash equities trading on NYSE MKT when it transitions to Pillar.

16. &thinspAt this time, the Exchange is not proposing rules, comparable to those in NYSE Arca Equities Rule 2, that specify the requirements to be approved as a member of the Exchange. Accordingly, the Exchange proposes that the rule numbers under Rule 2E that would support membership requirements would be designated as &ldquoReserved.&rdquo Instead, the Exchange's current rules governing the definition of a member organization and the requirements to be approved as a member organization would continue to apply.

17. &thinspNYSE Arca Equities Rule 3 Part I relates to board committees, which are described in the Exchange's Operating Agreement, which is available here:​publicdocs/​nyse/​regulation/​nyse-mkt/​Tenth_​Amended_​and_​Restated_​Operating_​Agreement_​of_​NYSE_​MKT_​LLC.pdf. NYSE Arca Equities Rules 3.4 and 3.5 relate to the self-regulatory responsibilities of NYSE Arca for the administration and enforcement of rules governing the operation of NYSE Arca Equities, its wholly owned subsidiary, and the delegation of authority from NYSE Arca to NYSE Arca Equities. Because the Exchange is itself a self-regulatory organization, these rules are inapplicable. The subject matter of NYSE Arca Equities Rule 3 Part III is addressed in the Exchange's Disciplinary Rules and Rule 2B&mdashEquities.

18. &thinspSee Securities Exchange Act Release No. 77679 (April 21, 2016), 81 FR 24908 (April 27, 2016) (File No. 4-631) (Order approving 10th Amendment to the LULD Plan).

19. &thinspSee Securities Exchange Act Release No. 79688 (December 23, 2016), 81 FR 96534 (December 30, 2016) (SR-NYSEArca-2016-170) (Notice of Filing and Immediate Effectiveness of Proposed Rule Change).

20. &thinspSee also infra proposed Rules 7.33E (Capacity Codes) and 7.41E (Clearance and Settlement).

21. &thinspSee also infra proposed Rule 7.36E regarding the display of orders on the Pillar trading platform.

22. &thinspSee supra note 10. The Exchange will file an amendment to the ETP Listing Rules Filing to add rule text for proposed paragraphs (b) and (c) of Rule 7.18E that would be based on NYSE Arca Equities Rule 7.18(b) and (c).

23. &thinspAs described in greater detail below, the Exchange proposes that the entirety of Rule 1000&mdashEquities would not be applicable to trading on the Pillar trading platform.

24. &thinspSee Securities Exchange Act Release No. 79705 (December 29, 2016), 82 FR 1419 (January 5, 2017) (SR-NYSEArca-2016-169) (Notice of Filing and Immediate Effectiveness of Proposed Rule Change).

25. &thinspAs described below, because the Exchange would not have Floor-based DMMs or trading, the remainder of Rule 116&mdashEquities would not be applicable to trading on the Pillar trading platform.

26. &thinspAs described below, the Exchange proposes that Rule 79A in its entirety would not be applicable on the Pillar trading platform.

27. &thinspSee Securities Exchange Act Release No. 77930 (May 26, 2016), 81 FR 35410 (June 2, 2016) (SR-NYSE-2016-38) (Notice of Filing and Immediate Effectiveness of Proposed Rule Change).

28. &thinspThe subject matter of Rule 17(a)&mdashEquities would be addressed in proposed Rule 13.2E. On Pillar, the Exchange would not operate with vendors and therefore would not need a vendor liability rule, as described in Rule 17(b)&mdashEquities. Current Rule 17(c)&mdashEquities would not be applicable because it addresses the same subject matter as proposed Rule 7.45E.

29. &thinspNYSE Arca Equities Rule 7.39 addresses the adjustment of open orders, e.g., orders with a good until canceled time-in-force instruction, due to corporate actions. Because the Exchange does not propose to have any open orders when trading on the Pillar trading platform, the Exchange will not adopt rule text based on NYSE Arca Equities Rule 7.39.

30. &thinspSee Rules 16&mdashEquities 20&mdashEquities 21&mdashEquities (Disqualification of Directors on Listing of Securities) Rule 26&mdashEquities (Disqualification of Directors on Listing of Securities) Rule 29&mdashEquities&mdashRule 34&mdashEquities Rule 38&mdashEquities&mdashRule 44&mdashEquities Rule 45&mdashEquities (Equities) Rule 50&mdashEquities Rule 57&mdashEquities&mdashRule 59&mdashEquities Rule 60A&mdashEquities Rule 65&mdashEquities Rule 69&mdashEquities Rule 92&mdashEquities Rule 106&mdashEquities Rule 107&mdashEquities Rule 109&mdashEquities&mdashRule 111&mdashEquities Rule 115&mdashEquities Rule 118&mdashEquities Rule 123G&mdashEquities Rule 124&mdashEquities Rule 132A&mdashEquities Rule 132B&mdashEquities Rule 132C&mdashEquities Rule 305&mdashEquities&mdash307&mdashEquities Rule 309&mdashEquities Rules 314&mdashEquities&mdash318&mdashEquities Rule 319&mdashEquities Rule 322&mdashEquities Rules 323&mdashEquities&mdash324&mdashEquities Rule 325&mdashEquities Rule 326(a)&mdashEquities Rule 326(b)&mdashEquities Rule 326(c)&mdashEquities Rule 326(d)&mdashEquities Rule 327&mdashEquities Rule 328&mdashEquities Rule 329&mdashEquities Rule 343&mdashEquities Rule 440A&mdashEquities and Rule 1003&mdashEquities.

33. &thinspSee Securities Exchange Act Release No. 15533 (January 29, 1979) (regarding the Amex Post Execution Reporting System, the Amex Switching System, the lntermarket Trading System, the Multiple Dealer Trading Facility of the Cincinnati Stock Exchange, the PCX's Communications and Execution System (&ldquoCOM EX&rdquo), and the Phlx's Automated Communications and Execution System (&ldquoPACE&rdquo)) (&ldquo1979 Release&rdquo).

34. &thinspSecurities Exchange Act Release Nos. 53128 (January 13, 2006) 71 FR 3550 (January 23, 2006) (File No. 10-13 1) (order approving Nasdaq Exchange registration) 58375 (August 18, 2008) 73 FR 49498 (August 21, 2008) (order approving BATS Exchange registration) 61152 (December 10, 2009) 74 FR 66699 (December 16, 2009) (order approving C2 exchange registration) and 78101 (June 17, 2016), 81 FR 41142, 41164 (June 23, 2016) (order approving Investors Exchange LLC registration).


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