Limit of a function at a point and at infinity

$\displaystyle\lim_{x\to x_0} f(x)=L$ — the “arbitrarily close” idea

A finite limit at $x_0$ : $f(x)$ settles stably to a number $L$ as $x$ is taken arbitrarily close to $x_0$ (often $x_0$ need not lie in $D(f)$ — only closeness within the set where the function is already defined matters). One-sided limits $x\to x_0^-$ , $x\to x_0^+$ approach from the left or right. The (two-sided) limit equals

L

if and only if both one-sided limits exist and equal $L$ . Limits at infinity: $x\to+\infty$ or $x\to-\infty$ — the same idea, but “closeness to $\infty$ ” means making

x

arbitrarily large in magnitude in the required direction.

Continuity at $x_0$ : $\displaystyle\lim_{x\to x_0} f(x)=f(x_0)$ and the point must lie in

D(f)

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