If f(x) is a differentiable function, then _ x a af(x) - xf(a) x … - Vidyadip

MCQ

If $f(x)$ is a differentiable function, then $\mathop {\lim }\limits_{x \to a} {{af(x) - xf(a)} \over {x - a}}$ is

✓
$af'\,(a) - f\,(a)$
B
$af\,(a) - f'(a)$
C
$af'\,(a) + f\,(a)$
D
$af\,(a) + f'(a)$

✓

Answer

Correct option: A.

$af'\,(a) - f\,(a)$

a
(a) $\mathop {\lim }\limits_{x \to a} \,\frac{{af(x) - xf(a)}}{{x - a}}$

==> $\mathop {\lim }\limits_{x \to a} \,\frac{{af(x) - xf(a) + af(a) - af(a)}}{{x - a}}$

==> $\mathop {\lim }\limits_{x \to a} \,\frac{{a[f(x) - f(a)] - f(a)[x - a]}}{{x - a}}$

==> $\mathop {\lim }\limits_{x \to a} \,\frac{{a[f(x) - f(a)]}}{{x - a}} - \mathop {\lim }\limits_{x \to a} \,f(a)$

==> $af'(a) - f(a)$.

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