12.4 Limits at Infinity and Limits of Sequences

A limit at infinity can be used to solve the area problem!

What a limit at infinity looks like: 

The limits as f(x) approaches the vertical asymptote x=2 is less important to what we are learning.

The limits as x approaches positive and negative infinity are what concerns us. 

As you can see, the limit of f(x) as x approaches negative infinity is 3. The limit of f(x) as x approaches infinity is 3. These limits mean that the value of f(x) gets arbitrarily close to 3 as x decreases or increases without bound. The limits give us the horizontal asymptotes to the left and to the right. 

Side-note: The graph of a rational function need not have a horizontal asymptote. If it does, however, its left and and right asymptotes must be the same. 

Something to keep in mind when evaluating limits at infinity:

When 

• If  r  is a positive real number, then the limit as x approaches infinity is 0. There is a limit toward the right.

• If   x^r  is defined when  x < 0,  then the limit as x approaches negative infinity is 0. There is a limit toward the left. 

Evaluating a limit at infinity: 

Examples:

1. Find the limit:

                                    

                                   

                                   

2. When finding the limit as x approaches infinity for a function such as

You can begin by dividing both the numerator and denominator by the highest-powered term in the denominator, in this case x^2.

In this case the limit is 0.

General Rules:

• When the degree of the numerator is less than the degree of the denominator, the limit is 0.
• When the degree of the numerator and denominator are equal, the limit is the ratio of the coefficients of the highest-powered terms. 
• When the degree of the numerator is  greater than the degree of the denominator, the limit does not exist.

If you are more of a visual and audio learner this video explains what I have just stated and shows some very helpful examples. 

Limits of sequences:

Limits of sequences have many of the same properties as limits of functions.

The limit of a sequence as it approaches infinity asks what the terms of the sequence get closer and closer to, or converge towards, as n increases without bound. 

A sequence that does not converse, for instance a sequence that oscillates, is said to diverge.

Let's say that:

and we are evaluating:

If   is a sequence such that  for every positive integer n

Then

Examples:

Here is a helpful video that shows how to evaluate the limit of a sequence:

I hoped this helped you with evaluating limits at infinity and evaluating the limit of a sequence!

GOOD LUCK TO ALL MY HONORS PRECALC BROTHERS AND SISTERS ON THIS UPCOMING FINAL.

STAY POSITIVE.

WORK HARD.

FOR WILHELM!  (AND BECAUSE WE ALL LOVE MATHS SO MUCH)

SHOUT OUT TO MR. WILHELM FOR BEING THE BEST MATH TEACHER OF ALL TIME. 

#legendary 

Thank you, 

Leah