Question

If function f(x) is differentiable at x = a, then $\underset{x\to a}{\lim }\frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$ is :

• A. $-a^2{f}^{\prime }(a)$
• B. $af(a)-a^2{f}^{\prime }(a)$
• C. $2af(a)-a^2{f}^{\prime }(a)0$
• D. $2af(a)+a^2{f}^{\prime }(a)$

Answer 1

Answer: (C)

$\underset{x\to a}{\lim }\frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$=\underset{x\to a}{\lim }\frac{2xf\left(a\right)-a^2{f}^{\prime }\left(x\right)}{1}$
$=2af\left(a\right)-a^2{f}^{\prime }\left(a\right)$
Alter$\underset{x\to a}{\lim }\frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$=\underset{x\to a}{\lim }\frac{x^{2}f\left(a\right)-a^{2}f\left(a\right)+a^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$=\underset{x\to a}{\lim }\frac{\left(x^2-a^2\right)f\left(a\right)+a^2\left(f\left(x\right)-f\left(a\right)\right)}{x-a}$
$=\underset{x\to a}{\lim }\left(x+a\right)f\left(a\right)-a^2\left\{\frac{f\left(x\right)-f\left(a\right)}{\left(x-a\right)}\right\}$
$=2af\left(a\right)-a^2{f}^{\prime }\left(a\right)$