Question

If f(x) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k is ______.

Options

• 1

• – 2

• 2

• `1/2`

Answer 1

If f(x) = `{{:((sin5x)/(x^2 + 2x)",", x ≠ 0),(k + 1/2",", x = 0):}` is continuous at x = 0, then the value of k is 2.

Explanation:

LHL = `lim_(h rightarrow 0) f(0 - h)`

= `lim_(h rightarrow 0) (sin5(0 - h))/((0 - h)^2 + 2(0 - h))`

= `-lim_(h -> 0) ((sin5h)/(5h))/(1/5(h - 2)) = 5/2`

RHL = `lim_(x rightarrow 0^+) (sin 5x)/(x^2 + 2x)`

= `lim_(x rightarrow 0^+) (sin5x)/(5x). lim_(x -> 0^+) 1/((x + 2)) = 5/2`

f(0) = `k + 1/2`

Since, it is continuous at x = 0

∴ LHL = RHL = f(0)

`\implies 5/2 = k + 1/2`

`\implies` k = 2