Limit at Infinity
Calculus & Advanced MathA 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: if you keep making $x$ bigger and bigger (or more and more negative), what number does the function's output home in on?
Example
For $f(x) = 1/x$, as $x$ gets huge ($100$, $1000$, $1{,}000{,}000\ldots$), the output ($0.01$, $0.001$, $0.000001\ldots$) shrinks toward $0$. So $\lim_{x \to \infty} 1/x = 0$.
Key Insight
Limits at infinity describe the long-run behavior of a function, like asking where a trend is ultimately headed.
Definition
$\lim_{x \to \infty} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ increases without bound. If the limit equals a finite $L$, the graph has a horizontal asymptote at $y = L$. Limits can also equal $+\infty$ or $-\infty$.
Example
$\lim_{x \to \infty} (3x^2 + 2)/(x^2 - 1)$: divide numerator and denominator by $x^2$ to get $(3 + 2/x^2)/(1 - 1/x^2) \to 3/1 = 3$. Horizontal asymptote: $y = 3$.
Key Insight
For rational functions, compare the degrees of numerator and denominator. Equal degrees give a finite limit equal to the ratio of leading coefficients.
Definition
$\lim_{x \to \infty} f(x) = L$ (epsilon-N definition): for every $\epsilon > 0$, there exists $N$ such that $x > N$ implies $|f(x) - L| < \epsilon$. This notion of limit at infinity is central to the study of asymptotic analysis and series convergence.
Example
$\lim_{x \to \infty} (\sin x)/x = 0$ by the squeeze theorem: $-1/x \le (\sin x)/x \le 1/x$ and both bounds $\to 0$. This result underlies the sinc function in signal processing.
Key Insight
L'Hopital's rule, the squeeze theorem, and asymptotic expansions are the principal tools for evaluating limits at infinity in analysis and applied mathematics.