Limit of a function
High school
Consider a function $f$ and a number $a$, and let $x$ get close to, but not equal to, $a$. The limit of $f(x)$ as $x$ approaches $a$ is the number that $f(x)$ gets close to.
The limit of $f$ is written as
\[\lim_{x \to a} f(x).\]
Undergraduate
Consider a function $f$ defined near $a$, except possibly at $a$. We write
\[\lim_{x \to a} f(x) = L\]and call $L$ the limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } 0 < \abs{x - a} < \delta.\]Examples
Worked examples
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$f(x) = \frac{\sin x}{x}$, at $x = 0$
: $\lim_{x \to 0} f(x) = 1$
High school
$f(0)$ is undefined, but the limit exists anyway.
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The step function: $0$ for $x < 0$ and $1$ for $x \geq 0$
: $\lim_{x \to 0} f(x)$ does not exist
Undergraduate
The left and right limits differ.
Properties
- The limit at $a$ exists if and only if the left limit and the right limit at $a$ both exist and are equal.
- The value $f(a)$, and whether it is defined, does not affect the limit at $a$. The function $f$ is continuous at $a$ when the limit equals $f(a)$.