Answer
(a) $ \dfrac{2}{5}$
(b) $ \dfrac{2}{5}$
Work Step by Step
(a) Since
\begin{align*}
\lim_{x\to \infty } f(x)&=\lim_{x\to \infty } \frac{2x+3}{5x+7}\\
&=\lim_{x\to \infty } \frac{2x/x+3/x}{5x/x+7/x}\\
&=\lim_{x\to \infty } \frac{2 +3/x}{5 +7/x}\\
&=\frac{2 +\lim_{x\to \infty }(3/x)}{5 +\lim_{x\to \infty }(7/x)}\\
&=\frac{2}{5}
\end{align*}
(b) Since
\begin{align*}
\lim_{x\to- \infty } f(x)&=\lim_{x\to \infty } \frac{2x+3}{5x+7}\\
&=\lim_{x\to -\infty } \frac{2x/x+3/x}{5x/x+7/x}\\
&=\lim_{x\to -\infty } \frac{2 +3/x}{5 +7/x}\\
&=\frac{2 +\lim_{x\to -\infty }(3/x)}{5 +\lim_{x\to- \infty }(7/x)}\\
&=\frac{2}{5}
\end{align*}
