For the function $f(x) = \left\{ \begin{array}{r} \frac{x^{3} - a^{3}}{x - a},\ x
eq a \ b,\ \ \ \ \ \ \ \ \ x = a \end{array} ight.\$, if $f(x)$ is continuous at $x = a$, then $b$ is equal to

Question

For the function $f(x) = \left\{ \begin{array}{r} \frac{x^{3} - a^{3}}{x - a},\ x 
eq a \ b,\ \ \ \ \ \ \ \ \ x = a \end{array} ight.\$, if $f(x)$ is continuous at $x = a$, then $b$ is equal to

Dr. Abubacker · Accepted Answer

(c)
LHL$= \lim_{x ightarrow a^{-}}\frac{x^{3} - a^{3}}{x - a} = \lim_{h ightarrow 0}\frac{(a - h)^{3} - a^{3}}{a - h - a}$
$= \lim_{h ightarrow 0}\frac{{(a - h - a)\{(a - h)}^{2} + a^{3} + a(a - h)\}}{- h} = 3a^{2}$
Since, $f(x)$ is continuous at $x = a$
$	herefore\ \ \ \ LHL = f(a)$
$\Rightarrow \ \ \ \ 3a^{2} = b$

Q. For the function $f(x) = \left\{ \begin{array}{r} \frac{x^{3} - a^{3}}{x - a},\ x \neq a \\ b,\ \ \ \ \ \ \ \ \ x = a \end{array} \right.\$, if $f(x)$ is continuous at $x = a$, then $b$ is equal to

Explanation

Reviewed by Dr. Abubacker

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