Question

If $f\left(x\right)=\left\{\begin{array}{cc}\frac{x^3-a^3}{x-a},& x\ne a\\ b,& x=a\end{array}\right.$ is continuous at x = a, then b = ______________.

Answer 1

It is given that, the function $f\left(x\right)=\left\{\begin{array}{cc}\frac{x^3-a^3}{x-a},& x\ne a\\ b,& x=a\end{array}\right.$is continuous at x = a.

$\therefore f\left(a\right)=\underset{x\to a}{\lim }f\left(x\right)$

$\Rightarrow b=\underset{x\to a}{\lim }\frac{x^3-a^3}{x-a}$

$\Rightarrow b=3a^2\left(\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}=n{a}^{n-1}\right)$


If $f\left(x\right)=\left\{\begin{array}{cc}\frac{x^3-a^3}{x-a},& x\ne a\\ b,& x=a\end{array}\right.$ is continuous at x = a, then b = $\underline{3a^2}$.