limit at infinity

The value a function's outputs approach as the input itself grows without bound.

We write $\displaystyle\lim_{x\to\infty} f(x) = L$ when $f(x)$ gets arbitrarily close to $L$ as $x$ grows without bound; the analogous statement holds as $x \to -\infty$ . A finite limit at infinity corresponds to a horizontal asymptote $y=L$ .

Examples

$\displaystyle\lim_{x\to\infty} \dfrac{3x+1}{x} = 3$ , so $y=3$ is a horizontal asymptote.