# Calculus 2 : Limits and Continuity

## Example Questions

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### Example Question #1 : Limits And Continuity

Find the values of  and  so that  is everywhere differentiable.

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 #1 : Limits And Continuity

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

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 #3 : Limits And Continuity

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

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

exists for any real number

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

may not be well defined for all real numbers.

All of the others are true.

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 #2 : Limits And Continuity

Consider the piecewise function:

What is ?

Limit does not exist.

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 #3 : Limits And Continuity

Consider the function .

I.

II.

III.

I and III

III only

I and II

II only

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 #6 : Limits And Continuity

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

at

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; function is not defined at the given value.

Function is continuous

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

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 #7 : Limits And Continuity

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

None of the above

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 #8 : Limits And Continuity

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

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 #9 : Limits And Continuity

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

None of the above

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 #1 : Limits And Continuity

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

None of the above

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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