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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 $b=\underline{3 a^{2}}$.