A) 3
B) 4
C) 5
D) 6
Correct Answer: C
Solution :
\[f'(1)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1+h)-f(1)}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1+h)}{h}-\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1)}{h}\] Since, \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1+h)}{h}=5\] \[\therefore \] \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1)}{h}\]will be finite. Since,\[f'(1)\]exist and\[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1)}{h}\]is finite. If\[f(1)=0\]and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1)}{h}=0\] \[\therefore \] \[f'(1)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1+h)}{h}=5\]
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