Right limit
High school
Consider a function $f$ and a number $a$, and let $x$ approach $a$ from above (from the right). The right limit of $f$ at $a$, written $\lim_{x \to a^{+}} f(x)$, is the number that $f(x)$ approaches.
Undergraduate
Consider a function $f$ defined on an interval to the right of $a$. We write $\lim_{x \to a^{+}} f(x) = L$, and call $L$ the right limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } a < x < a + \delta.\]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
$\frac{\sin x}{x}$ approaches $1$ from both sides of $0$, so the left limit, the right limit, and the limit are all $1$, even though $f(0)$ is undefined.
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The step function: $0$ for $x < 0$ and $1$ for $x \geq 0$
: $\lim_{x \to 0^{+}} f(x) = 1$
Undergraduate
The step function jumps at $0$, so its left limit is $0$ and its right limit is $1$.