Limit at Infinity
A limit at infinity describes what value a function approaches as its input grows without bound in the positive or negative direction.
Formula
\lim_{x \to \infty} f(x) = L
Definition
A limit at infinity asks what number a function's output homes in on as you keep making $x$ bigger and bigger (or more and more negative). Formally, $\lim_{x \to \infty} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ increases without bound: for every $\epsilon > 0$ there exists $N$ such that $x > N$ implies $|f(x) - L| < \epsilon$. If the limit equals a finite $L$, the graph has a horizontal asymptote at $y = L$; limits can also equal $+\infty$ or $-\infty$.
Example
For $f(x) = 1/x$, as $x$ gets huge ($100$, $1000$, $1{,}000{,}000\ldots$), the output shrinks toward $0$, so $\lim_{x \to \infty} 1/x = 0$. For rational functions like $\lim_{x \to \infty} (3x^2 + 2)/(x^2 - 1)$, dividing numerator and denominator by $x^2$ gives $(3 + 2/x^2)/(1 - 1/x^2) \to 3/1 = 3$, a horizontal asymptote at $y=3$. A more subtle case, $\lim_{x \to \infty} (\sin x)/x = 0$, follows from the squeeze theorem since $-1/x \le (\sin x)/x \le 1/x$ and both bounds go to $0$, a result underlying the sinc function in signal processing.
Key Insight
Limits at infinity describe the long-run behavior of a function, like asking where a trend is ultimately headed; for rational functions, comparing the degrees of numerator and denominator (equal degrees give a finite limit equal to the ratio of leading coefficients) is the quickest way to evaluate them. L'Hopital's rule, the squeeze theorem, and asymptotic expansions are the principal tools for evaluating limits at infinity in analysis and applied mathematics.