A) \[a\,\,f'(a)-f(a)\]
B) \[a\,\,f(a)-f'(a)\]
C) \[a\,\,f'(a)+f(a)\]
D) \[\,f'(a)+a\,f(a)\]
Correct Answer: A
Solution :
\[\underset{x\to a}{\mathop{\lim }}\,\,\frac{a\,f(x)-x\,f\,(a)}{x-a}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{a\,f(x)-a\,f(a)+a\,f(a)-x\,f(a)}{x-a}\]\[=\underset{x\to a}{\mathop{\lim }}\,\frac{a[f(x)-f(a)]}{(x-a)}-\underset{x\to 0}{\mathop{\lim }}\,\,f(a)\] \[=a\,f'\,(a)-f(a)\]
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