if `f(x)=(e^(x^(2))-cosx)/(x^(2))`, for `xne0` is continuous at `x=0`, then value of f(0) is

Options

A. `(2)/(3)`
B. `(5)/(2)`
C. `1`
D. `(3)/(2)`

Answer

D

Explanation

We have,
`f(x)=(e^(x^(2))-cosx)/(x^(2))` is continous at x=0
`therefore underset(xto0)(lim)f(x)=f(0)`
`impliesf(0)=underset(xto0)(lim)(e^(x^(2))-cosx)/(x^(2))`
`impliesf(0)=underset(xto0)(lim)(2x e^(x^(2))+sinx)/(2x)` [apply L' Hospital rule]
`impliesf(0)=underset(xto0)(lim)(2x e^(x^(2)))/(2x)+underset(x to 0)(lim)(sinx)/(2x)+1+(1)/(2)=(3)/(2)`