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))-cos x )/(x^(2))` is continuous at x = 0
`:.underset(x rarr 0) lim f(x) = f(0)`
`rArr f(0)= underset(x rarr0)lim (e^(x^(2))-cos x)/(x^(2))`
`rArr f(0) = underset(x rarr 0) lim (2xe^(x^(2))+sin x)/(2x)` [ apply L'Hospital rule ]
`rArr f(0) = underset(x rarr 0) lim (2xe^(x^(2)))/(2x)+ underset( x rarr 0) lim (sin x)/(2x) = 1+ 1/2 = 3/2`