Determine one-sided limits

A one-sided limit describes the value a function approaches as $x$ gets arbitrarily close to $a$ from only one direction: $x<a$ for a left-hand limit and $x>a$ for a right-hand limit, regardless of whether $f(a)$ is defined or equals that value. These limits can be determined from graphs, tables, and algebraic forms, including finite behavior and divergence to $+\infty$ or $-\infty$ ; a two-sided limit exists only when the corresponding one-sided limits agree. More abstract generalized limits and advanced theoretical edge cases are not included.

Detailed Explanation: 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 $x=2$ .

Learn by doing: Determine one-sided limits

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Limits - One-Sided Table with Discontinuity (Words) to One-Sided Limit

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