Let `f: R->R` be such that `f(1)=3` and `f^(prime)(1)=6.` Then `lim_(x->0)((f(1+x))/(f(1)))^(1//x)=` (a)`1` (b) `e^(1/2)` (c) `e^2` (d) `e^3`

Options

A. 1
B. `e^(1//2)`
C. `e^2`
D. `e^3`

Answer

C

Explanation

We have
`lim_(xto0) {(f(1+x))/(f(1))}^(1//x)[1^oo"form"]`
` =lim_(xto0) {1+(f(1+x)-f(1))/(f(1))}^(1//x)`
` =e^(lim_(xto0) (f(1+x)-f(1))/(xf(1))`
` =e^((1)/(f(1))lim lim_(xto0) (f(x+1)-f(1))/(x)`
` =e^((f'(1))/(f(1)))=e^(6//3)=e^2[because lim_(xto0)(f(1+x)-f(1))/(x)=f'(1)]`