Answer
When $\epsilon = 0.5,~~~$ let $~~~\delta = 0.2$
When $\epsilon = 0.1,~~~$ let $~~~\delta = 0.04$
Work Step by Step
$f(x) = \frac{e^{2x}-1}{x}$
Let $\epsilon = 0.5$
$f(-0.2) = \frac{e^{2(-0.2)}-1}{-0.2} = 1.65$
$f(0.2) = \frac{e^{2(0.2)}-1}{0.2} = 2.46$
Let $\delta = 0.2$
Then:
If $0 \lt \vert x-0 \vert \lt \delta,~~$ then $~~0 \lt \vert f(x)-2 \vert \lt \epsilon$
Let $\epsilon = 0.1$
$f(-0.04) = \frac{e^{2(-0.02)}-1}{-0.02} = 1.92$
$f(0.04) = \frac{e^{2(0.04)}-1}{0.04} = 2.08$
Let $\delta = 0.04$
Then:
If $0 \lt \vert x-0 \vert \lt \delta,~~$ then $~~0 \lt \vert f(x)-2 \vert \lt \epsilon$
