Limits at Infinity

Infinite limits were about the output blowing up near a number. Limits at infinity flip the question: let the input grow without bound and ask what does. The answer is a precise version of the end behavior you met in algebra, and when the function levels off it gives you the horizontal asymptote. One idea runs through the whole lesson: as gets huge, the biggest term wins.

What Is a Limit at Infinity?

We write when gets as close to as you like once is large enough. The same idea with describes the left end of the graph. Either way, the line is a horizontal asymptote, the limit-based version of the definition in the horizontal asymptotes lesson.

The most important basic fact is

A fixed number divided by something enormous is tiny. The same is true as , as long as is defined for negative .

Rules for Limits Involving Infinity

The limit laws still apply as . When a piece heads to infinity, though, some combinations have a definite answer and others don’t.

The indeterminate forms are where the work is. could be anything: , and all start out as . You have to rewrite the expression to find out which.

How to Find a Limit at Infinity: Divide by the Highest Power

For a rational function, divide every term on the top and bottom by the highest power of in the denominator. Then every term except the leading ones turns into something like , which disappears.

Find . Divide by :

A table shows the same thing happening:

Each extra zero on adds roughly one more correct digit of . The lower-degree terms and matter less and less, which is exactly why dividing by and letting them vanish gives the right answer. The same idea explains why only the leading terms control end behavior for polynomials.

Comparing Degrees

Doing that division in general gives a shortcut. If the top has degree and the bottom has degree :

• : the bottom grows faster, so the limit is 0. Example: .
• : the limit is the ratio of the leading coefficients, as in the example.
• : the top wins and the function grows without bound. This is an infinite limit at infinity: , because the expression behaves like .

For , the first two cases are the same. In the third, check the sign of the leading behaviour: behaves like , so it goes to on the left.

Limits at Infinity With Square Roots

Find and the same limit as .

Divide by , and inside the root by . For , , so

For , though, , which flips the sign: the limit as is . This function has two different horizontal asymptotes, on the right and on the left. Square roots are the usual reason a graph has different behaviour at each end.

Infinity Minus Infinity

Find .

Both terms grow without bound, so this is . Don’t guess 0. Multiply by the conjugate, as in evaluating limits:

Now divide top and bottom by : . The two infinities don’t cancel; they differ by an amount that settles at 3.

Exponentials and Other Functions

• and : exponential decay gives the asymptote .
• : the inverse tangent levels off at two horizontal asymptotes, .
• Growth rates: as , grows slower than any power of , and any power of grows slower than . So and . L’Hôpital’s rule, at the end of this chapter, proves these.
• has no limit at infinity; it keeps oscillating. But by the squeeze theorem.

Common Mistakes to Avoid

• Treating as 0 or as 1. Both are indeterminate.
• Forgetting . As , pulling out of a square root introduces a minus sign.
• Dividing by the wrong power. Use the highest power in the denominator.
• Confusing the two kinds of limit. is a limit at infinity; is an infinite limit.

Practice Problems

Work each one before opening the answer.

Problem 1. Find .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. Find the horizontal asymptotes of .

Next, continuity: what it means for a function to have no breaks, and the three conditions that make it precise.