Limits (Graph)

One-Sided Limits

Definition. We say the limit of $f(x)$ is equal to $L$ as $x$ approaches $c$ from the left and write $\lim_{x\to c^-}f(x) = L$ if we can make values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $c$ with $x$ less than $c$ .

Example 1. Evaluate the following limits based on the graph.

Evaluate $\displaystyle\lim_{x\to 2^-} f(x)$
Evaluate $\displaystyle\lim_{x\to 0^-} f(x)$
Evaluate $\displaystyle\lim_{x\to (-1)^-} f(x)$

Definition. We say the limit of $f(x)$ is equal to $L$ as $x$ approaches $c$ from the right and write $\lim_{x\to c^+}f(x) = L$ if we can make values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $c$ with $x$ greater than $c$ .

Example 2. Evaluate the following limits based on the graph.

Evaluate $\displaystyle\lim_{x\to 2^+} f(x)$
Evaluate $\displaystyle\lim_{x\to 0^+} f(x)$
Evaluate $\displaystyle\lim_{x\to (-1)^+} f(x)$

Two-Sided Limits

Definition. Suppose $f(x)$ is defined on some open interval that contains $c$ , except possibly at $c$ itself. Then we write $\lim_{x\to c}f(x) = L$ and say “the limit of $f(x)$ equals $L$ , as $x$ approaches $c$ ” if we can make values of $f(x)$ arbitrarily close to $L$ by restricting $x$ to be sufficiently close to $c$ (on either side of $c$ ) but not equal to $c$ .

$\lim_{x\to c}f(x) = L \Longleftrightarrow \lim_{x\to c^-}f(x) = \lim_{x\to c^+}f(x) = L$

Example 3. Evaluate the following limits based on the graph.

Evaluate $\displaystyle\lim_{x\to 2} f(x)$
Evaluate $\displaystyle\lim_{x\to 0} f(x)$
Evaluate $\displaystyle\lim_{x\to (-1)} f(x)$

Practice Problems.

Evaluate the following limits based on the given graph.

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f = plot(-x, -3, -1)
f += point([-1, 1], color = "white", markeredgecolor = "blue", size = 20)
​
f += plot(cos(3 * x), -1, 1)
​
f += plot(sqrt(x), 1, 3)
f += point([1, 1], color = "white", markeredgecolor = "blue", size = 20)
show(f)

$\displaystyle\lim_{x\to (-1)^-} f(x)=$
$\displaystyle\lim_{x\to (-1)^+} f(x)=$
$\displaystyle\lim_{x\to (-1)} f(x)=$
$f(-1)=$

$\displaystyle\lim_{x\to 0^-} f(x)=$
$\displaystyle\lim_{x\to 0^+} f(x)=$
$\displaystyle\lim_{x\to 0} f(x)=$
$f(0)=$

$\displaystyle\lim_{x\to 1^-} f(x)=$
$\displaystyle\lim_{x\to 1^+} f(x)=$
$\displaystyle\lim_{x\to 1} f(x)=$
$f(1)=$

Limit Involves Infinity

Example 4. Evaluate the following limits based on the graph.

$\displaystyle\lim_{x\to -\infty} f(x)=$
$\displaystyle\lim_{x\to \infty} f(x)=$
$\displaystyle\lim_{x\to (-1)^+} f(x)=$
$\displaystyle\lim_{x\to (-1)^-} f(x)=$
$\displaystyle\lim_{x\to 1^+} f(x)=$
$\displaystyle\lim_{x\to 1^-} f(x)=$