Question

Let `f(x) = (2 - sqrt(x + 4))/(sin 2x), x ≠ 0`. In order that f(x) is continuous at x = 0, f(0) is to be defined as ______.

Options

• `(-1)/8`

• `1/2`

• 1

• `1/8`

Answer 1

Let `f(x) = (2 - sqrt(x + 4))/(sin 2x), x ≠ 0`. In order that f(x) is continuous at x = 0, f(0) is to be defined as `bbunderline((-1)/8)`.

Explanation:

`f(0) = lim_(x → 0) f(x)`

= `lim_(x → 0) (2 - sqrt(x + 4))/(sin 2x)`   [0/0 form]

= `lim_(x → 0) (-1/(2sqrt(x + 4)))/(2 cos 2x)`  [Using L’ Hospital rule]

= `lim_(x → 0) (-1)/(sqrt(x + 4) cos 2x)`

 = `(-1)/(4 xx sqrt4 xx 1)`

= `(-1)/8`