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  4. It is given that f'(a) exists, then displaystyle limx → a (xf (a)-af (x)/(x-a)) is equal to

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Q. It is given that $f'(a)$ exists, then $\displaystyle \lim_{x \to a} $ $\frac{xf \left(a\right)-af \left(x\right)}{(x-a)}$ is equal to

Continuity and Differentiability

Solution:

$\displaystyle \lim_{x \to a} $ $\frac{xf \left(a\right)-af \left(x\right)}{x-a}$
$=\displaystyle \lim_{x \to a} $ $\frac{xf \left(a\right)-xf \left(x\right)+xf \left(x\right)-af \left(x\right)}{\left(x-a\right)}$
$=\displaystyle \lim_{x \to a} $ $\left[\frac{x\left[f \left(a\right)-f \left(x\right)\right]}{\left(x-a\right)}+\frac{f \left(x\right)\left(x-a\right)}{\left(x-a\right)}\right]$
$=\displaystyle \lim_{x \to a} $ $-\frac{x\left[f \left(x\right)-f \left(a\right)\right]}{x-a}+$$\displaystyle \lim_{x \to a} f (x)$
$= -a f'(a)+ f(a) $
$= f(a)-af'(a)$