1.15 Connecting Limits at Infinity and Horizontal Asymptotes

A different idea is a limit at infinity, which asks what happens as x becomes very large positive or very large negative, such as

\lim_{x \to \infty} f(x)

\lim_{x \to -\infty} f(x)

For polynomials, limits at infinity are determined entirely by the leading term, because the highest power dominates all lower-degree terms for very large x. So for a polynomial like

a_nx^n+\cdots

you mainly look at

a_nx^n

if the degree is even, both ends go the same direction; if the leading coefficient is positive, both go to

\infty

and if negative, both go to

-\infty

If the degree is odd, the ends go opposite directions; with a positive leading coefficient,

x \to \infty

gives

\infty

and

x \to -\infty

gives

-\infty

while a negative leading coefficient reverses that. For rational functions, compare degrees: if the numerator’s degree is smaller, the limit is 0; if the degrees are equal, the limit is the ratio of leading coefficients; if the numerator’s degree is larger, the function grows without bound or behaves like a polynomial based on dominant terms. So the short rule is: for polynomials, use the leading term; for rational functions, compare top and bottom degrees and leading terms.

Example 1: Find

\lim_{x \to \infty} (3x^4-2x+7)

The leading term is 3x⁴, which has even degree and positive coefficient, so

\lim_{x \to \infty} (3x^4-2x+7)=\infty

Example 2: Find

\lim_{x \to \infty} \frac{3x+1}{x^2-4}

The numerator has lower degree than the denominator, so

\lim_{x \to \infty} \frac{3x+1}{x^2-4}=0

Example 3: Find

\lim_{x \to \infty} \frac{4x^2-1}{2x^2+3}

The numerator and denominator have the same degree, so use the ratio of leading coefficients:

\lim_{x \to \infty} \frac{4x^2-1}{2x^2+3}=\frac{4}{2}=2

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