A) \[1\]
B) \[{{e}^{1/2}}\]
C) \[{{e}^{2}}\]
D) \[{{e}^{3}}\]
Correct Answer: C
Solution :
Let\[y={{\left[ \frac{f(1+x)}{f(1)} \right]}^{1/x}}\] \[\Rightarrow \] \[\log y=\frac{1}{x}[\log f(1+x)-\log f(1)]\] \[\Rightarrow \]\[\underset{x\to 0}{\mathop{\lim }}\,\log y=\underset{x\to 0}{\mathop{\lim }}\,\frac{[\log f(1+x)-\log f(1)]}{x}\] \[\Rightarrow \]\[\underset{x\to 0}{\mathop{\lim }}\,\log y=\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{1}{f(1+x)}f(1+x) \right]\] (using L Hospitals rule) \[\Rightarrow \] \[\underset{x\to 0}{\mathop{\lim }}\,y={{e}^{2}}\]
You need to login to perform this action.
You will be redirected in
3 sec
