Left limit
High school
Consider a function $f$ and a number $a$, and let $x$ approach $a$ from below (from the left). The left 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 left of $a$. We write $\lim_{x \to a^{-}} f(x) = L$, and call $L$ the left limit of $f$ at $a$, if for every $\varepsilon > 0$ there is a $\delta > 0$ such that
\[\abs{f(x) - L} < \varepsilon \quad \text{whenever } a - \delta < x < a.\]Examples
Worked examples
- The step function: $0$ for $x < 0$ and $1$ for $x \geq 0$ : $\lim_{x \to 0^{-}} f(x) = 0$ Undergraduate