Calculus 2 : Limits and Continuity

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Limits And Continuity

Find the values of  and  so that  is everywhere differentiable.

Possible Answers:

Correct answer:

Explanation:

If a function is to be everywhere differentiable, then it must also be continuous everywhere. This implies that the one sided limits must be equal: .

Next, the one sided limits of the derivative must also be equal. That is, .

Now we have two missing variables and two equations. Set up a system and solve by elimination or substitution. 

Example Question #496 : Limits

Rational_graph

The graph above is a sketch of the function . For what intervals is  continuous?

Possible Answers:

Correct answer:

Explanation:

 For a function to be continuous at a point must exist and

This is true for all values of  except and .

Therefore, the interval of continuity is .

Example Question #2 : Limits And Continuity

Suppose that  are continuous functions on . Which of the following is FALSE?

Possible Answers:

 may not be well defined for all real numbers.

 exists for any real number

 has either an absolute maximum, absolute minimum, or both.

 is well defined for all real numbers, and is continuous on .

All of the others are true.

Correct answer:

 has either an absolute maximum, absolute minimum, or both.

Explanation:

In order to show this statement is false, we must provide one counterexample. 

For example, let . Both of these are continuous functions (Their graphes are connected with no "jumps" or "breaks"). Then , but  does not have an absolute maximum, or mimimum.

Example Question #498 : Limits

Consider the piecewise function:  

What is ?

Possible Answers:

Limit does not exist.

Correct answer:

Limit does not exist.

Explanation:

The piecewise function

 

indicates that  is one when  is less than five, and is zero if the variable is greater than five.  At , there is a hole at the end of the split.  

The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right.  From the left to right, the limit approaches 1 as  approaches negative five.   From the right, the limit approaches zero as  approaches negative five.

Since the limits do not coincide, the limit does not exist for .

Example Question #499 : Limits

Consider the function .

Which of the following statements are true about this function? 

I.

II. 

III. 

Possible Answers:

II only

I and II

III only

I and III

Correct answer:

I and II

Explanation:

For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point. 

 First, notice that 

This means that the function is continuous everywhere.

Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.

 

Thus, the limit as x approaches three exists and is equal to , so I and II are true statements. 

Example Question #3 : Limits And Continuity

Determine whether or not the function is continuous at the given value of .

 at 

Possible Answers:

Function is NOT continuous; the limit of the function does not exist and function is not defined at the given value.

Function is NOT continuous; the limit of the function does not exist.

Function is NOT continuous; function is not defined at the given value.

Function is continuous

Correct answer:

Function is continuous

Explanation:

The definition of continuity at a point is

.

Since  is in the interval of the domain of the function, we see 

Also, since the function approaches  as  approaches  from the left and the right, we see that

.

Since 

the function is continuous at .

Example Question #4 : Limits And Continuity

Screen shot 2015 08 17 at 6.27.41 pm

Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

None of the above

Correct answer:

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for . Thus,  is continuous on the interval .

Example Question #5 : Limits And Continuity

Screen shot 2015 08 17 at 6.36.23 pm

Given the above graph of , over which of the following intervals is  continuous?

 

Possible Answers:

Correct answer:

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for  and . Thus,  is continuous on the interval .

Example Question #2 : Limits And Continuity

Screen shot 2015 08 17 at 6.45.07 pm

Given the above graph of , over which of the following intervals is  continuous?

Possible Answers:

None of the above

Correct answer:

Explanation:

For a function  to be continuous at a given point , it must meet the following two conditions:

1.) The point  must exist, and

2.) .

 

Examining the above graph,  is continuous at every possible value of  except for  and . Thus,  is continuous on the interval .

Example Question #7 : Limits And Continuity

Screen shot 2015 08 18 at 9.51.12 am

Given the above graph of , over which of the following intervals is continuous?

 

Possible Answers:

None of the above

Correct answer:

Explanation:

For a function to be continuous at a given point , it must meet the following two conditions:

1.) The point must exist, and

2.) .

 

Examining the above graph, is continuous at every possible value of except for . Thus, is continuous on the interval .

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