Question

If f(x) = `{{:(log(sec^2 x)^(cot^2x)",", "for"  x ≠ 0),(K",", "for"  x = 0):}`

is continuous at x = 0, then K is ______.

Options

• e^–1

• 1

• e

• 0

Answer 1

If f(x) = `{{:(log(sec^2 x)^(cot^2x)",", "for"  x ≠ 0),(K",", "for"  x = 0):}`

is continuous at x = 0, then K is 1.

Explanation:

We have, f(x) = `{{:(log(sec^2 x)^(cot^2x)",", "for"  x ≠ 0),(k",", "for"  x = 0):}`

As, f(x) is continuous at x = 0

So, f(0) = `lim_(x rightarrow 0)[log(sec^2x)^(cot^2x)]`

= `lim_(x rightarrow 0)[cot^2x log(sec^2x)]`

= `lim_(x rightarrow 0) (log(1 + tan^2x))/(tan^2x)`

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

= 1