Precalculus : Limits

Example Questions

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 #1 : Limits

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 : Limits

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 #4 : 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 #5 : 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 #1 : Finding Limits As X Approaches Infinity

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)?

Nothing can be concluded from the given function.

The speed of the car approaches a constant number.

The speed of the car approaches zero.

The speed of the car depends on the starting speed.

The speed of the car approaches infinity.

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 #6 : 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 .

.

Example Question #11 : Limits

The Michaelis-Menten equation is important in chemical kinetics. Suppose we are given the following equation relating K (reaction rate) and C (concentration):

Determine:

Limit Does Not Exist

Explanation:

There are a number of ways to solve this. Either we can solve by finding what K(C) evaluates to for larger values of C and see where they converge, i.e. :

K(10000) = 952.38

K(1000000) = 999.5

etc...

And we see that K(C) approaches 1000 for larger concentrations, C.

Or we can notice that the dominant term in the numerator is 1000C; dominant term in the denominator is C.

1000C / C = 1000, which will ultimately be our limit.

Example Question #12 : Limits

Let .

Find  .

Undefined

Explanation:

To find    here, you need only plug in  for :

Evaluate

.