# 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

does not exist because

does not exist because

exists 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 #2 : Using Limits With Continuity

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

exists because

does not exist because

does not exist because

does not exist because  is not continuaous at .

does not exist 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 : Limits

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?

does not exist because  is not continuous at .

does not exist because .

exists because  is constant on .

does not exist because  is undefined.

exists because .

exists because .

Explanation:

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

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