#### Answer

$(3.99,4.01)$
$\delta=0.01$

#### Work Step by Step

We are given $f(x)=x+1$, $L=5$, $c=4$, and $\epsilon=0.01$.
To find the desired interval about $c$, we plug the above into $|f(x)-L| \lt \epsilon$ and solve for $x$:
$$\left|x+1-5 \right| \lt 0.01\\ -0.01 \lt x+1-5 \lt 0.01\\ -0.01 \lt x-4 \lt 0.01\\
3.99 \lt x \lt 4.01.$$
Thus our interval is $(3.99,4.01)$.
Now, to find our $\delta$, we note that
$$3.99 \lt x \lt 4.01 \implies -0.01 \lt x-4 \lt 0.01.$$
Hence for $\delta=0.01$,
$$0 \lt |x-4| \lt \delta \implies |f(x)-5| \lt \epsilon.$$