Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 56

Suppose $\displaystyle \lim \limits_{x \to 1} \frac{f(x)}{x^2} = 5$, Find the following limits:
$\displaystyle \text{(a) } \lim \limits_{x \to 0} f(x) \qquad \text{(b) } \lim \limits_{x \to 0} \frac{f(x)}{x}$






$
\begin{equation}
\begin{aligned}

\text{(a) } & \lim \limits_{x \to 0} \frac{f(x)}{x^2} = 5
\qquad
\Longrightarrow
\qquad
\frac{\lim \limits_{x \to 0} f(x) }{ \lim \limits_{x \to 0} x^2} = 5
\qquad
\text{ (Multiplying $\lim \limits_{x \to 0} x^2$ to both sides of the equator) }\\

\phantom{x} & \frac{\lim \limits_{x \to 0} f(x) }{\cancel{ \lim \limits_{x \to 0} x^2}}
\cdot
\cancel{ \lim \limits_{x \to 0} x^2}
= 5 \cdot \lim \limits_{x \to 0} (x^2)\\

\phantom{x} & \lim \limits_{x \to 0} f(x) = 5 \cdot \lim \limits_{x \to 0} x^2\\
\phantom{x} & \lim \limits_{x \to 0} f(x) = 5 \cdot (0)^2 = 0\\
\phantom{x} & \lim \limits_{x \to 0} f(x) = 0\\

\text{(b) } & \lim \limits_{x \to 0} \frac{f(x)}{x^2} = 5
\qquad
\Longrightarrow
\qquad
\frac{\lim \limits_{x \to 0} f(x)}{\lim \limits_{x \to 0} x \cdot \lim \limits_{x \to 0} x} = 5
\qquad
\text{ (Multiplying $\lim \limits_{x \to 0} x$ to both sides of the equation)}\\

\phantom{x} & \frac{\lim \limits_{x \to 0} f(x)}{\lim \limits_{x \to 0} x \cdot \cancel{\lim \limits_{x \to 0} x}}
\cdot
\cancel{\lim \limits_{x \to 0} x }
= 5 \cdot \lim \limits_{x \to 0} x\\

\phantom{x} & \lim \limits_{x \to 0} \frac{f(x)}{x} = 5 \cdot \lim \limits_{x \to 0} x\\
\phantom{x} & \lim \limits_{x \to 0} \frac{f(x)}{x} = 5 \cdot 0\\
\phantom{x} & \lim \limits_{x \to 0} \frac{f(x)}{x} = 0

\end{aligned}
\end{equation}
$