# Precalculus : Find the Limit of a Function

## Example Questions

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

Evaluate the limit below:

1

0

Explanation:

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 #2 : Find The Limit Of A Function

Find the limit

Explanation:

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 #4 : Limits

Solve the following limit:

Explanation:

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

### Example Question #3 : Find The Limit Of A Function

Evaluate the following limit.

Explanation:

The function has a removable discontinuity at  .  Once a factor of  is "divided out" the resultant function is , which evaluates to  as  approaches 0.

### Example Question #4 : Find The Limit Of A Function

Let .

Find .

The limit does not exist.

Explanation:

This is a graph of . We know that  is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as  approaches 0 from the left,  approaches negative infinity.

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.

is actually infinity, not negative infinity.

### Example Question #5 : Find The Limit Of A Function

Calculate .

The limit does not exist.

Explanation:

This can be rewritten as follows:

We can substitute , noting that as ,

, which is the correct choice.

### Example Question #6 : Find The Limit Of A Function

Find the limit as x approaches infinity

Explanation:

As x approaches infinity we only need to look at the highest order of polynomial in both the numerator and denominator. Then we compare the highest order polynomial in both the numerator and denominator. If the denominator is higher order our limit goes to zero, if the numerator is higher our order our limit goes to positive or negative infinity (depending on the sign of the highest order x term). If our numerator and denominator have the same order the limit goes to a/b where a is the coefficient for the highest order x in the numerator and b is the coefficient for the highest order x in the denominator.

Our numerator has higher order and the coefficient for the x to the fourth term is negative so our limit goes to negative infinity.

### Example Question #7 : Find The Limit Of A Function

Solve the limit as  approaches infinity.

Explanation:

As x approaches infinity we only need to look at the highest order of polynomial in both the numerator and denominator. Then we compare the highest order polynomial in both the numerator and denominator. If the denominator is higher order our limit goes to zero, if the numerator is higher our order our limit goes to positive or negative infinity (depending on the sign of the highest order x term). If our numerator and denominator have the same order the limit goes to a/b where a is the coefficient for the highest order x in the numerator and b is the coefficient for the highest order x in the denominator.

For our equation the orders are the same in x for the numerator and denominator (both 4). So we divide the coefficients of the highest order x terms to get our limit and we get

### Example Question #8 : Find The Limit Of A Function

The speed of a car traveling on the highway is given by the following function of time:

What can you say about the car's speed after a long time (that is, as  approaches infinity)?

The speed of the car approaches a constant number.

Nothing can be concluded from the given function.

The speed of the car depends on the starting speed.

The speed of the car approaches infinity.

The speed of the car approaches zero.

The speed of the car approaches infinity.

Explanation:

The function given is a polynomial with a term , such that  is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as  approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

### Example Question #9 : Find The Limit Of A Function

Finding limits of rational functions.

Let

.

Find

.

Undefined

Explanation:

First factor the numerator to simplify the function.

,

so

.

Now

.

There is no denominator now, and hence no discontinuity.  The limit can be found by simply plugging in  for .

.

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