A) \[-{{a}^{2}}f'(a)\]
B) \[af(a)-{{a}^{2}}f\,'(a)\]
C) \[2af(a)-{{a}^{2}}f\,'(a)0\]
D) \[2af(a)+{{a}^{2}}f'(a)\]
Correct Answer: C
Solution :
\[\underset{x\to a}{\mathop{\ell im}}\,\frac{{{x}^{2}}f(a)-{{a}^{2}}f(x)}{x-a}\] \[=\underset{x\to a}{\mathop{\ell im}}\,\frac{2xf(a)-{{a}^{2}}f'(x)}{1}\] \[=2af(a)-{{a}^{2}}f'(a)\] Alter\[\underset{x\to a}{\mathop{\ell im}}\,\frac{{{x}^{2}}f(a)-{{a}^{2}}f(x)}{x-a}\] \[=\underset{x\to a}{\mathop{\ell im}}\,\frac{{{x}^{2}}f(a)-{{a}^{2}}f(a)+{{a}^{2}}f(a)-{{a}^{2}}f(x)}{x-a}\] \[=\underset{x\to a}{\mathop{\ell im}}\,\frac{({{x}^{2}}-{{a}^{2}})f(a)-{{a}^{2}}f(x)-f((a))}{x-a}\] \[=\underset{x\to a}{\mathop{\ell im}}\,(x+a)f(a)-{{a}^{2}}\left\{ \frac{f(x)-f(a)}{(x-a)} \right\}\] \[=2af(a)-{{a}^{2}}f'(a)\]
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