Question

If f(x) = `(e^(x^2) - cos x)/x^2`, for x ≠ 0 is continuous at x = 0, then value of (0) is ______.

Options

• `2/3`

• `5/2`

• 1

• `3/2`

Answer 1

If f(x) = `(e^(x^2) - cos x)/x^2`, for x ≠ 0 is continuous at x = 0, then value of (0) is `underlinebb(3/2)`.

Explanation:

f(x) = `(e^(x^2) - cos x)/x^2`

Since f(x) is continuous at x = 0

`implies` f(0) = `lim_(x rightarrow 0) ` f(x)

`implies` f(0) = `lim_(x rightarrow 0) ((e^(x^2) - 1) - (cosx - 1))/x^2`

`implies` f(0) = `lim_(x rightarrow 0) (e^(x^2) - 1)/x^2 - lim_(x rightarrow 0) (-2sin^2  x/2)/x^2`

`implies` f(0) = `1 + 2 lim_(x rightarrow 0) [(sin  x/2)/(x/2)]^2 xx 1/4 = 1 + 2/4 = 3/2`