If `f(x)` is differentiable and strictly increasing function, then the value of `("lim")_(xvec0)(f(x^2)-f(x))/(f(x)-f(0))` is 1 (b) 0 (c) `-1` (d) 2

Options

A. 1
B. 0
C. `-1`
D. 2

Answer

C

Explanation

We have, `underset(x to 0)lim (f(x^(2))-f(x))/(f(x)-f(0))=underset(x to 0)lim (f(x)^(2)-f(0)-(f(x)-f(0)))/(f(x)-f(0))`
`underset(x to 0)lim (f(x^(2))-f(x))/(f(x)-f(0))=underset(x to 0)lim {(f(x)^(2)-f(0))/(f(x)-f(0))-1}`
`underset(x to 0)lim (f(x^(2))-f(x))/(f(x)-f(0))=underset(x to 0)lim (f(x)^(2)-f(0))/(x^(2)-0)xx(x^(2)-0)/(f(x)-f(0))-1`
`underset(x to 0)lim (f(x^(2))-f(x))/(f(x)-f(0))=underset(x to 0)lim (f(x)^(2)-f(0))/(x^(2))xx ((x)/(f(x)-f(0)))/(x-0)-1`
`underset(x to 0)lim (f(x^(2))-f(x))/(f(x)-f(0))=f'(0)xx(0)/(f'(0))xx(0)/(f(0))-1=-1 [{:(,therefore f'(x) gt 0),(,"for all x"):}]`