Answer
$\delta=\dfrac{9}{25}$
Work Step by Step
We see from the graph that in order for $f(x)$ to be within $\epsilon=0.5$ of $L=2$, we must have
$$-\dfrac{16}{9} \lt x \lt -\dfrac{16}{25}.$$
Subtracting $c=-1$ from all three sides gives
$$-\dfrac{7}{9} \lt x-(-1) \lt \dfrac{9}{25}.$$
Note that
$$-\dfrac{9}{25} \lt x-(-1) \lt \dfrac{9}{25} \implies -\dfrac{7}{9} \lt x-(-1) \lt \dfrac{9}{25}.$$
Hence for $\delta = \dfrac{9}{25}$,
$$0 \lt |x-(-1)| \lt \delta \implies 0 \lt |f(x)-2| \lt \epsilon.$$
