### All High School Math Resources

## Example Questions

### Example Question #1 : Using Limits With Continuity

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

**Possible Answers:**

does not exist because .

exists because

does not exist because

does not exist because

exists because

**Correct answer:**

does not exist because .

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?

**Possible Answers:**

does not exist because

does not exist because

does not exist because

does not exist because is not continuaous at .

exists because

**Correct answer:**

exists because

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 #3 : Using Limits With Continuity

A function is defined by the following piecewise equation:

At , the function is:

**Possible Answers:**

discontinuous

continuous

**Correct answer:**

continuous

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 #4 : Using Limits With Continuity

The graph depicts a function . Does exist?

**Possible Answers:**

does not exist because is not continuous at .

exists because is constant on .

exists because .

does not exist because is undefined.

does not exist because .

**Correct answer:**

exists because .

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

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