1.3 One-Sided Limits
Notes
A one-sided limit looks at only one direction.
- $\displaystyle \lim_{x \to a^-} f(x)$ is the left-hand limit (inputs smaller than $a$).
- $\displaystyle \lim_{x \to a^+} f(x)$ is the right-hand limit (inputs larger than $a$).
- $\displaystyle \lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.
- Example: $f(x) = x + 1$ for $x < 2$ and $f(x) = x^2$ for $x \geq 2$. The left limit at 2 is 3 and the right limit is 4, so the limit does not exist.
Practice
Let $f(x) = x + 1$ for $x < 2$ and $f(x) = x^2$ for $x \geq 2$. Find $\displaystyle \lim_{x \to 2^-} f(x)$.
For inputs below 2 use $x + 1$, which approaches 3.
For the function ($x + 1$ for $x < 2$, $x^2$ for $x \geq 2$), what is $\displaystyle \lim_{x \to 2} f(x)$?
The left limit is 3 and the right limit is 4. They differ.
Find $\displaystyle \lim_{x \to 0^-} \frac{|x|}{x}$.
For $x < 0$, $|x| = -x$, so the fraction equals $-1$.
Ready for more? Take the Unit 1 test.