The Limit of a Function

Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.

One-Sided Limits

Sometimes, what happens to us at a place depends on the direction we use to approach that place. If we approach Niagara Falls from the upstream side, then we will be 182 feet higher and have different worries than if we approach from the downstream side. Similarly, the values of a function near a point may depend on the direction we use to approach that point. If we let $x$ approach 3 from the left ( $x$ is close to $\mathrm{3}$ and $x < 3)$ , then the values of $[x] \operatorname{INT}(x)$ equal $\mathrm{2}$ (Fig. 7). If we let $x$ approach 3 from the right ( $x$ is close to 3 and $x>3$ ), then the values of $[\mathrm{x}]=\operatorname{INT}(\mathrm{x})$ equal $\mathrm{3}$ .

On the number line we can approach a point from the left or right, and that leads to one-sided limits.

Definition of Left and Right Limits:

The left limit as $x$ approaches $c$ of $f(x)$ is $L$ if the values of $f(x)$ get as close to $L$ as we want when $x$ is very close to and left of $\mathbf{c}, x < \mathbf{c}: \lim\limits_{x \rightarrow c^{-}} f(x)=L$ .

The right limit, written with $\mathbf{x} \rightarrow \mathbf{c}^{+}$ , requires that $\mathrm{x}$ lie to the right of $\mathbf{c}, \mathrm{x} > \mathbf{c}$ .

Example 5: Evaluate

$\lim\limits_{x \rightarrow 2^{-}}(x-[x])$ and

$\lim\limits_{x \rightarrow 2^{+}}(x-[x])$ .

Solution: $\quad$ The left-limit notation $\mathrm{x} \rightarrow 2^{-}$ requires that $\mathrm{x}$ be close to 2 and that $x$ be to the left of 2, so $x < 2$ .

If $1 < x < 2$ , then $[x]=1$ so $\quad \lim\limits_{x \rightarrow 2^{-}}(x-[x])=2-1=1$ .

If $x$ is close to 2 and is to the right of 2, then $2 < x < 3$ so $[x]=2$ and $\lim _{x \rightarrow 2^{+}}(x-[x])=2-2=0$ .

The graph of $f(x)=x-[x]$ is shown in Fig. 8.

If the left and right limits have the same value, $\lim\limits_{x \rightarrow c^{-}} f(x)=\lim\limits_{x \rightarrow c^{+}} f(x)=L$ , then the value of $\mathrm{f}(\mathrm{x})$ is close to $\mathrm{L}$ whenever $\mathrm{x}$ is close to $\mathrm{c}$ , and it does not matter if $\mathrm{x}$ is left or right of $\mathrm{c}$ so $\lim\limits_{x \rightarrow c} f(x)=L$ . Similarly, if $\lim\limits_{x \rightarrow c} f(x)=L$ , then $\mathrm{f}(\mathrm{x})$ is close to $\mathrm{L}$ whenever $\mathrm{x}$ is close to $\mathrm{c}$ and less than $\mathrm{c}$ and whenever $\mathrm{x}$ is close to $\mathrm{c}$ and greater than $\mathrm{c}$ , so $\lim\limits_{x \rightarrow c^{-}} f(x)=\lim\limits_{x \rightarrow c^{+}} f(x)=L$ . We can combine these two statements into a single theorem.

One-Sided Limit Theorem:

$\lim\limits_{x \rightarrow c} f(x)=L \quad$ if and only if $\lim\limits_{x \rightarrow c^{-}} f(x)=\lim\limits_{x \rightarrow c^{+}} f(x)=L$ .

Corollary:

If $\lim\limits_{x \rightarrow c^{-}} f(x) \neq \lim\limits_{x \rightarrow c^{+}} f(x)$ , then $\lim\limits_{x \rightarrow c} f(x)$ does not exist.

One-sided limits are particularly useful for describing the behavior of functions which have steps or jumps.

To determine the limit of a function involving the greatest integer or absolute value or a multiline definition, definitely consider both the left and right limits.

Practice 3: Use the graph in Fig. 9 to evaluate the one and two-sided limits of $\mathrm{f}$ at $\mathrm{x}=0,1,2$ , and $3$ .

Practice 4: Let $f(x)= \begin{cases}1 & \text { if } x < 1 \\ x & \text { if } 1 < x < 3 \\ 2 & \text { if } 3 < x\end{cases}$

Find the one and two-sided limits of $\mathrm{f}$ at $\mathrm{1}$ and $\mathrm{3}$ .