Skill: Determine one-sided limits

A one-sided limit tells you what value $f(x)$ approaches as $x$ gets close to $a$ from only one direction:

• From the left: $\displaystyle \lim_{x\to a^-}f(x)$ means $x<a$.
• From the right: $\displaystyle \lim_{x\to a^+}f(x)$ means $x>a$.

Worked example

Suppose

$$f(x)= \begin{cases} x+1, & x<2,\\ 7-x, & x\ge 2. \end{cases}$$

Find the one-sided limits as $x\to 2$.

1. Find the left-hand limit

For values of $x$ less than $2$, use the rule $f(x)=x+1$:

$$\lim_{x\to 2^-}f(x) =\lim_{x\to 2^-}(x+1) =2+1 =3.$$

So, as $x$ approaches $2$ from the left, $f(x)$ approaches $3$.

2. Find the right-hand limit

For values of $x$ greater than $2$, use the rule $f(x)=7-x$:

$$\lim_{x\to 2^+}f(x) =\lim_{x\to 2^+}(7-x) =7-2 =5.$$

So, as $x$ approaches $2$ from the right, $f(x)$ approaches $5$.

3. Compare the one-sided limits

$$\lim_{x\to 2^-}f(x)=3 \qquad\text{and}\qquad \lim_{x\to 2^+}f(x)=5.$$

Because the one-sided limits are different,

$$\lim_{x\to 2}f(x)$$

does not exist.

The actual value $f(2)$ does not change either one-sided limit. One-sided limits depend on the values the function approaches near $2$, not necessarily on the value at $x=2$.