Question

If f(x) = `{{:(cos ((π(sqrt(1 + x) - 1))/x)/x,",", x ≠ 0),(π/k,",", x = 0):}`

is continuous at x = 0, then k² is equal to ______.

Options

• 61.00

• 62.00

• 63.00

• 64.00

Answer 1

If f(x) = `{{:(cos ((π(sqrt(1 + x) - 1))/x)/x,",", x ≠ 0),(π/k,",", x = 0):}`

is continuous at x = 0, then k² is equal to 64.00.

Explanation:

`π/k = lim_(x→0) cos(π/(sqrt(1 + x) + 1))/x`

⇒ `π/k = lim_(x→0) (sin(π/2 - π/(sqrt(1 + x) + 1)))/x`

⇒ `π/k = lim_(x→0) (sin(π/2 ((sqrt(1 + x) - 1))/((sqrt(1 + x) + 1))))/x` 

⇒ `π/k = lim_(x→0) (sin((πx)/(2(sqrt(1 + x) + 1)^2)))/x`

k² = 64

k = 8