Limits at Infinity

Limits at Infinity: How to Approach Them

When a problem asks for a limit as (x) goes to infinity or negative infinity, you are studying the long-term behavior of the expression. The goal is to find what value the function approaches, or whether it grows without bound.

1) Identify the dominant terms

For rational expressions, compare the highest powers of (x) in the numerator and denominator. Lower-degree terms become less important as (x) gets very large in magnitude.

2) Simplify by the largest power

A reliable method is to divide every term by the highest power of (x) appearing in the denominator. This often makes the limiting behavior clear because terms like (1/x) or (1/x^2) go to (0).

3) Evaluate the remaining expression

After simplification, take the limit term by term. Typical outcomes include:

• a finite number,
• (0),
• or an unbounded result such as (\infty) or (-\infty) when the function does not level off.

4) Check your answer

Make sure the result matches the growth rates of the main terms. For example, if the numerator has lower degree than the denominator, the limit should be (0). If the degrees are equal, the limit is usually the ratio of the leading coefficients.

Common caution

Do not substitute large numbers directly unless the expression is simple; algebraic simplification is usually safer and more exact.

Quick self-check

Ask: Which terms survive when all lower-order terms vanish? That surviving behavior determines the limit.