# High School Math : Using Limits with Continuity

## Example Questions

### Example Question #1 : Using Limits With Continuity The above graph depicts a function . Does exist, and why or why not? exists because  exists because  does not exist because . does not exist because  does not exist because  does not exist because .

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

### Example Question #1 : Limits The above graph depicts a function . Does exist, and why or why not? does not exist because  does not exist because  does not exist because  does not exist because is not continuaous at . exists because  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 : Calculus Ii — Integrals

A function is defined by the following piecewise equation: At , the function is:

discontinuous

continuous

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 #2 : Using Limits With Continuity The graph depicts a function . Does exist? exists because is constant on . does not exist because . does not exist because is not continuous at . does not exist because is undefined. exists because . exists because . exists if and only if ; the actual value of is irrelevant.
As can be seen, and ; therefore, , and exists. 