Lesson Explainer: Limits at Infinity Mathematics • Higher Education

In this explainer, we will learn how to evaluate limits of a function when 𝑥 tends to infinity.

By definition, we will say that lim𝑓(𝑥)=𝐿, where 𝐿 is a real number provided that the values 𝑓(𝑥) get as close as we like to 𝐿 and that 𝑥 is big enough.

Likewise, lim𝑓(𝑥)=𝐿 means that 𝑓(𝑥)𝐿 for all 𝑥 that are large enough but negative.

Before looking at examples of convergence, let us note that there are essentially two things that can go wrong with convergence as 𝑥 goes to (or ):

  • The function oscillates too much, so there is no number 𝐿. This is the case with 𝑓(𝑥)=𝑥sin.
  • The values 𝑓(𝑥) increase (or decrease) without bound as the positive or negative value of 𝑥 gets larger and larger.

In the figure shown, we have the following:

(a) is the graph of 𝑓(𝑥)=𝑥 which goes to + as the positive value of 𝑥 gets larger, while it goes to as the negative value of 𝑥 gets larger. We write these as limandlim𝑥=+𝑥=.

(b) is the graph of the polynomial function 𝑝(𝑥)=143500(𝑥+10)(𝑥+1)(𝑥20)(𝑥22)(𝑥30) from which we see that, like 𝑓(𝑥)=𝑥, the values become infinite in absolute value as |𝑥| also becomes infinite, but in the opposite sense to 𝑓: limandlim𝑝(𝑥)=𝑝(𝑥)=+.

(c) is the graph of the exponential function 𝑔(𝑥)=2, where the graph suggests strongly that limandlim2=+2=0.

In fact, in order to determine the behavior of limits at infinity of the two classes polynomials and rational functions, it is enough to know the following:

  1. lim𝑥=+.
  2. lim𝑥=.
  3. Infinite limits behave with respect to sums and products as expected, so that, for example, if lim𝑓(𝑥)=+, and lim𝑔(𝑥)=+, then limlim𝑓(𝑥)+𝑔(𝑥)=+,𝑓(𝑥)𝑔(𝑥)=+. Differences and quotients are not as straightforward (see below). Corresponding facts hold for .
  4. The following is how infinite limits interact with constants. If lim𝑓(𝑥)=+, then limiflimif𝐴𝑓(𝑥)=+𝐴>0,𝐴𝑓(𝑥)=𝐴<0. For any real number 𝐴, limlim(𝑓(𝑥)+𝐴)=+𝐴𝑓(𝑥)=0. Corresponding facts hold for .
  5. Limits that would evaluate to or 0 or (whether + or ) are all indeterminate and require some algebraic simplification if they can be evaluated at all.

As a result, we have lim3𝑥=0 because this is the product of (3) and 0=1𝑥lim. In fact, since lim𝑥=, we see from property (3) that lim𝑥=, and then lim3𝑥=0 by property (4). We also have that lim2𝑥3𝑥70=, because 2𝑥3𝑥70=𝑥23𝑥70𝑥 which has limit ()×(2)= as 𝑥.

In order to determine the limit lim2𝑥3𝑥+1, we proceed, as with all rational functions, by dividing both the numerator and the denominator by the highest power of 𝑥 visible in both these polynomials: 2𝑥3𝑥+1=+=13+ so that limlimlimlim2𝑥3𝑥+1=13+=2𝑥13+1𝑥=013+0=13.

Example 1: Finding the Limit of a Polynomial Function at Infinity

Find lim2𝑥4𝑥3𝑥7.

Answer

We factor out 𝑥, the highest power of 𝑥: 2𝑥4𝑥3𝑥7=𝑥24𝑥3𝑥7𝑥.

Now lim𝑥=+ while lim24𝑥3𝑥7𝑥=2000=2.

Therefore, limlimlim2𝑥4𝑥3𝑥7=𝑥24𝑥3𝑥7𝑥=(+)(2)=+.

Notice that we choose to factor the polynomial rather that argue by summands. We could have started by saying that lim2𝑥=+, lim4𝑥=+, and lim3𝑥=+, but then what is 7?

And the answer is that we cannot decide, because is an indeterminate form.

Of course, the algebra merely affirms the intuition that the leading term determines where the polynomial goes as 𝑥 tends to infinity. The 2𝑥 term dominates when 𝑥 is large.

Example 2: Finding the Limit of a Rational Function at Infinity

Find lim5𝑥92𝑥+5.

Answer

Following the strategy for polynomials, factor out leading terms first, and then cancel those: 5𝑥92𝑥+5=(𝑥)5(𝑥)2+=(𝑥)5(𝑥)2+=5(𝑥)2+.

Now we can “substitute” for the limits: limlimlimlimlim5𝑥92𝑥+5=5𝑥2+=1𝑥×59𝑥2+5𝑥=0×52=0.

Next is an alternative strategy.

Example 3: Finding the Limit of a Rational Function at Infinity

Find lim𝑥7𝑥+3𝑥+7𝑥+48𝑥6𝑥6𝑥+4.

Answer

First, note that the numerator is a polynomial of degree 4 and the denominator is a polynomial of degree 3.

As the degree of the numerator is greater than the degree of the denominator, the limit will be infinite.

Since the leading coefficient in the numerator is negative (the coefficient of 𝑥), therefore the limit is .

And the following is another way.

Example 4: Finding the Limit of a Rational Function at Infinity

Find lim3𝑥𝑥𝑥+3𝑥+2𝑥8𝑥𝑥5𝑥8.

Answer

If 𝑛, and 𝑎 is a constant, then lim𝑎𝑥=0. Also, if 𝑐 is a constant, then lim𝑐=𝑐. Use these facts, along with the fact that dividing both the numerator and the denominator of a fraction by the same quantity results in an equivalent fraction, to find lim3𝑥𝑥𝑥+3𝑥+2𝑥8𝑥𝑥5𝑥8 as follows: limlimlimlimlimlimlimlimlimlimlimlim3𝑥𝑥𝑥+3𝑥+2𝑥8𝑥𝑥5𝑥8÷𝑥(𝑥)=3++1=31𝑥1𝑥+3𝑥+2𝑥18𝑥1𝑥5𝑥8𝑥=300+0+010000=3.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.