Skill: Evaluate limits from tables

A table helps you estimate $\lim_{x\to a} f(x)$ by looking at function values as $x$ gets close to $a$ from both sides.

Worked example

Use the table to estimate $\displaystyle \lim_{x\to 2} f(x)$.

$x$ values less than $2$	$f(x)$	$x$ values greater than $2$	$f(x)$
$1.9$	$3.9$	$2.1$	$4.1$
$1.99$	$3.99$	$2.01$	$4.01$
$1.999$	$3.999$	$2.001$	$4.001$

Suppose also that $f(2)=10$.

Step 1: Look from the left.

As $x$ approaches $2$ from values less than $2$, the function values approach $4$:

$$\lim_{x\to 2^-}f(x)=4.$$

Step 2: Look from the right.

As $x$ approaches $2$ from values greater than $2$, the function values also approach $4$:

$$\lim_{x\to 2^+}f(x)=4.$$

Step 3: Compare the two sides.

Because both one-sided limits approach the same number, the two-sided limit is

$$\boxed{\lim_{x\to 2}f(x)=4}.$$

Notice that the limit is $4$, even though $f(2)=10$. A limit describes the value that $f(x)$ approaches near $x=2$; it does not have to equal the actual value at $x=2$.

If the values from the left and right approached different numbers, then the two-sided limit would not exist.