The graph of $f(x)$ is shown (see figure).
$f(x) = \frac{6x^2}{x^2 + 7}$
(a) Find the following limit.
$L = \lim_{x \to \infty} f(x) = 6$
(b) Determine $x_1$ and $x_2$ in terms of $\epsilon$.
$x_1 = -\sqrt{\frac{42}{\epsilon} - 7}$
$x_2 = \sqrt{\frac{42}{\epsilon} - 7}$
(c) Determine $M$, where $M > 0$, such that $|f(x) - L| < \epsilon$ for $x > M$.
$M = \sqrt{\frac{42}{\epsilon} - 7}$
(d) Determine $N$, where $N < 0$, such that $|f(x) - L| < \epsilon$ for $x < N$.
$N = -\sqrt{\frac{42}{\epsilon} - 7}$