Question

If f(x) = `{{:((cosec^2x)^{tan^2x}","  x ≠ 0), (k","    x = 0):}` is continuous at x = 0, then k is equal to ______ 

Options

• 0

• 1

• e

• e²

Answer 1

If f(x) = `{{:((cosec^2x)^{tan^2x}","  x ≠ 0), (k","    x = 0):}` is continuous at x = 0, then k is equal to e. 

Explanation:

Since, f(x) is continuous at x = 0.

∴ f(0) = `lim_{x→0} f(x)`

⇒ k = `lim_{x→0}(cosec^2x)^{tan^2x}`

⇒ k = `lim_{x→0} (1 + cot^2x)^{1/(cos^2x)}`

⇒ k = e