Answer
$$\lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x} =3a^2$$
Work Step by Step
Given $$\lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x}$$
let $$ f(x) = \frac{(x+a)^{3}-a^{3}}{x} $$
Since, we have
$$ f(0)= \frac{a^3-a^{3}}{0}=\frac{0}{0}$$
So, transform algebraically and cancel
\begin{aligned}L&= \lim _{x \rightarrow 0} \frac{(x+a)^{3}-a^{3}}{x} \\
&= \lim _{x \rightarrow 0} \frac{(x+a-a)((x+a)^2+a(x+a)+a^2)}{x} \\
&= \lim _{x \rightarrow 0} \frac{(x )((x+a)^2+a(x+a)+a^2)}{x} \\
&= \lim _{x \rightarrow 0}( (x+a)^2+a(x+a)+a^2) \\
&=((0+a)^2+a(0+a)+a^2)\\
&=a^2+a^2+a^2\\
&=3a^2
\end{aligned}
