Question:
If $f(x)=\left\{\begin{array}{cl}\frac{x^{3}-a^{3}}{x-a}, & x \neq a \\ b, & x=a\end{array}\right.$ is continuous at $x=a$, then $b=$
Solution:
It is given that, the function $f(x)=\left\{\begin{array}{cl}\frac{x^{3}-a^{3}}{x-a}, & x \neq a \\ b, & x=a\end{array}\right.$ is continuous at $x=a$.
$\therefore f(a)=\lim _{x \rightarrow a} f(x)$
$\Rightarrow b=\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a}$
$\Rightarrow b=3 a^{2} \quad\left(\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}\right)$
If $f(x)=\left\{\begin{array}{cl}\frac{x^{3}-a^{3}}{x-a}, & x \neq a \\ b, & x=a\end{array}\right.$ is continuous at $x=a$, then $b=\underline{3 a^{2}}$.