Question

The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` is ______.

Options

• 1

• –1

• 0

• does not exist

Answer 1

The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` is 1.

Explanation:

`lim_(x rightarrow 0) (sqrt(1 + x^2) - sqrt(1 - x^2))/x^2.(sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) + sqrt(1 - x^2))`

= `lim_(x rightarrow 0) (1 + x^2 - 1 + x^2)/(x^2(sqrt(1 + x^2) + sqrt(1 - x^2))`

= `2/(sqrt(1) + sqrt(1))`

= `2/2`

= 1