You can often estimate a limit before you can compute it, from a table of values or from a graph. The key is to approach from both sides separately.
Key idea
The left-hand limit $\displaystyle\lim_{x \to a^-} f(x)$ uses $x < a$; the right-hand limit $\displaystyle\lim_{x \to a^+} f(x)$ uses $x > a$. The two-sided limit exists exactly when both one-sided limits exist and are equal, and then it is that common value. In a table, read the values as $x$ closes in on $a$ from each side; on a graph, follow the curve toward $x = a$ from each side and note the height it heads for.
Why it works
$x$ can approach $a$ from two directions. If the values head to $2$ from the left but to $3$ from the right, there is no single number they all get close to, so the two-sided limit does not exist. The dot at $x = a$ itself, if there is one, plays no part.
| $x$ | $-0.1$ | $-0.01$ | $-0.001$ | $0.001$ | $0.01$ | $0.1$ |
|---|---|---|---|---|---|---|
| $\frac{\sin x}{x}$ | $0.99833$ | $0.99998$ | $1.00000$ | $1.00000$ | $0.99998$ | $0.99833$ |
Worked examples
Worked example: From a table
Use the table above to estimate $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$.
- From the left ($-0.1$, $-0.01$, $-0.001$) the values climb toward $1$.
- From the right the values match, also approaching $1$.
Answer: $1$ (an important limit you will meet again in calculus)
Worked example: A jump
$f(x) = x + 1$ for $x < 1$ and $f(x) = x + 2$ for $x \ge 1$. Find both one-sided limits at $1$ and the two-sided limit.
- Left: $x + 1 \to 2$. Right: $x + 2 \to 3$.
- $2 \ne 3$.
Answer: $2$, $3$; the two-sided limit does not exist
Worked example: A hole with a stray dot
A graph follows the line $y = 4 - x$ on both sides of $x = 1$, with an open circle at $(1, 3)$ and a filled dot at $(1, 0)$. Find $\displaystyle\lim_{x \to 1} f(x)$ and $f(1)$.
- From both sides the curve heads to the open circle at height $3$.
- The filled dot says $f(1) = 0$, but that does not affect the limit.
Answer: The limit is $3$; $f(1) = 0$
Common mistake
Reading the filled dot instead of where the curve is heading, or checking only one side. Cover the point $x = a$ with your finger and trace toward it from the left, then from the right. If the two traces end at different heights, the limit does not exist.