### All Precalculus Resources

## Example Questions

### Example Question #1 : Limits

At what value of is the function discontinuous,

?

**Possible Answers:**

There is no hole

**Correct answer:**

To find discontinuity we need to look at where the denominator of the function is equal to zero. Looking at our function,

we need to set the denominator equal to zero and solve for :

When , which is undefined.

Therefore is where the function is discontinuous.

### Example Question #1 : Limits

What is the

?

**Possible Answers:**

The limit does not exist

**Correct answer:**

Means to find the limit of the function as approaches from the left. We can see that is a vertical asymptote therefore we need to look at numbers extremely close to zero on the left side. The value of continues to decrease to the left of zero, with the value of decreasing even more as gets closer to . Thus, the limit is

### Example Question #3 : Limits

What is the

?

**Possible Answers:**

The limit does not exist

**Correct answer:**

We first need to simplify the function, we can do this by factoring the numerator and denominator.

If we plug in 3 into the simplified function, we get:

### Example Question #2 : Limits

What is the

?

**Possible Answers:**

**Correct answer:**

Substituting in for we get the following:

### Example Question #1 : Introductory Calculus

Evaluate the following:

**Possible Answers:**

limit does not exist

**Correct answer:**

When evaluating limits at infinity there are three rules to keep in mind:

- If the degree of the highest exponent in the numerator is equal to the degree of the highest exponent in the denominator, then the limit is equal to the ratio of the coefficient of the highest exponent in the numerator over the coefficient of the highest exponent in the denominator. Make sure to include signs.
- If the degree of the highest exponent in the numerator is less than the degree of the highest exponent in the denominator, the limit = 0.
- If the degree of the highest exponent in the numerator is greater than the degree of the higest exponent in the denominator, divide the highest power in the numerator by the highest power in the denominator and substitute for inifity. You will either subsitute for positive or negative infinity based on what the questions asks you to evaluate the limit at.

In this case, the degree is higher in the numerator than the denominator (rule #3). Hence, you need to divide the highest powers and evaluate.

Evaluate as x:

Answer: limit =

### Example Question #2 : Limits

Evaluate the following:

**Possible Answers:**

**Correct answer:**

When evaluating limits at infinity there are three rules to keep in mind:

- If the degree of the highest exponent in the numerator is equal to the degree of the highest exponent in the denominator, then the limit is equal to the ratio of the coefficient of the highest exponent in the numerator over the coefficient of the highest exponent in the denominator. Make sure to include signs.
- If the degree of the highest exponent in the numerator is less than the degree of the highest exponent in the denominator, the limit = 0.
- If the degree of the highest exponent in the numerator is greater than the degree of the higest exponent in the denominator, divide the highest power in the numerator by the highest power in the denominator and substitute for inifity. You will either subsitute for positive or negative infinity based on what the questions asks you to evaluate the limit at.

In this case, the both the numerator and denominator have the highest degree of an exponent of 4 (rule #1). Hence, you need to compare the ratio of the coefficients.

Answer: limit =

### Example Question #1 : Limits

What is the,

?

**Possible Answers:**

**Correct answer:**

The end behavior of the function follows the highest powers in both the numerator and denominator. Therefore, to find the limit we need to look only at the term: as those are the highest powers in the numerator and demoninator.

Now we take the

.

Thus the limit of our original function is also .

### Example Question #2 : Limits

Evaluate the limit below:

**Possible Answers:**

0

1

**Correct answer:**

will approach when approaches , so will be of type as shown below:

So, we can apply the L’ Hospital's Rule:

since:

hence:

### Example Question #1 : Find The Limit Of A Function

Find the limit

**Possible Answers:**

**Correct answer:**

When x=3/2 our denominator is zero so we can't just plug in 3/2 to get our limit. If we look at the numerator when x=3/2 we find that it is zero as well so our numerator can be factored. We see that our limit can be re-written as:

we then can cancel the 2x-3 from the numerator and denominator leaving us with:

and we can just plug in 3/2 into this limit to get

note: our function is not continuous at x=3/2 but the limit does exist.

### Example Question #1 : Find The Limit Of A Function

Solve the following limit:

**Possible Answers:**

**Correct answer:**

To solve this problem we need to expand the term in the numerator

when we do that we get

the second degree x terms cancel and we get

now we can cancel our h's in the numerator and denominator to get

then we can just plug 0 in for h and we get our answer