High School Math : Using Limits with Continuity

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Using Limits With Continuity

Function

 

The above graph depicts a function . Does  exist, and why or why not?

Possible Answers:

 does not exist because 

 exists because 

 exists because 

 does not exist because 

 does not exist because .

Correct answer:

 does not exist because .

Explanation:

 exists if and only if . As can be seen from the diagram, , but . Since ,    does not exist.

Example Question #2 : Limits

Function

The above graph depicts a function . Does  exist, and why or why not?

Possible Answers:

 does not exist because  is not continuaous at .

 does not exist because 

 exists because 

 does not exist because 

 does not exist because 

Correct answer:

 exists because 

Explanation:

 exists if and only if ;

the actual value of  is irrelevant, as is whether  is continuous there.

As can be seen,

 and ;

therefore, ,

and  exists.

Example Question #2 : Using Limits With Continuity

A function is defined by the following piecewise equation:

At , the function is:

Possible Answers:

continuous

discontinuous

Correct answer:

continuous

Explanation:

The first step to determine continuity at a point is to determine if the function is defined at that point. When we substitute in 3 for , we get 18 as our -value. is thus defined for this function.

The next step is determine if the limit of the function is defined at that point. This means that the left-hand limit must be equal to the right-hand limit at . Substitution reveals the following:

Both sides of the function, therefore, approach a -value of 18.

 

Finally, we must ensure that the curve is smooth by checking the limit of the derivative of both sides.

 

Since the function passes all three tests, it is continuous.

Example Question #3 : Using Limits With Continuity

Function

The graph depicts a function . Does  exist?

Possible Answers:

 does not exist because  is not continuous at .

 exists because  is constant on .

 does not exist because .

 does not exist because  is undefined.

 exists because .

Correct answer:

 exists because .

Explanation:

 exists if and only if ; the actual value of  is irrelevant.

As can be seen,  and ; therefore, , and   exists.

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