Question

Find the indicated limit, if it exists. (2 points) limit of f of x as x approaches 5 where f of x equals 5 minus x when x is less than 5, 8 when x equals 5, and x plus 3 when x is greater than 5

Answer 1

It looks like we have

$f(x)=\begin{cases}5-x&\text{for }x5\end{cases}$

and we want to find $\lim\limits_{x\to5}f(x)$.

Since $x$ is approaching 5, we don't care about the value of $f(x)$ when $x=5$.

We do care about how $f(x)$ behaves to either side of $x=5$. If $x\to5$ from below, then $f(x)=5-x$, so that

$\displaystyle\lim_{x\to5^-}f(x)=\lim_{x\to5}(5-x)=5-5=0$

On the other hand, if $x\to5$ from above, then $f(x)=x+3$, so that

$\displaystyle\lim_{x\to5^+}f(x)=\lim_{x\to5}(x+3)=5+3=8$

The one-sided limits do not match, since 0 ≠ 8, so the limit does not exist.