Question

If the function f(x) defined by

f(x) = `{{:(x sin  1/x",", "for"  x = 0),(k",", "for"  x = 0):}`

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

Options

• 0

• 1

• –1

• `1/2`

Answer 1

If the function f(x) defined by

f(x) = `{{:(x sin  1/x",", "for"  x = 0),(k",", "for"  x = 0):}`

is continuous at x = 0, then k is equal to 0.

Explanation:

Given, f(x) = `{{:(x sin  1/x",", "for"  x = 0),(k",", "for"  x = 0):}`

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

So, LHL = RHL = f(0)  ..........(i)

Now, LHL = `lim_(x rightarrow 0^-) f(x) = lim f(0 - h)`

= `lim_(h rightarrow 0) (0 - h) sin  1/((0 - h))`

= `lim_(h rightarrow 0) (-h)sin(- 1/h)`

= `lim_(h rightarrow 0) hsin  1/h` = 0 × finite value lies between –1 and 1 = 0

`[∵ lim_(h rightarrow 0) sin  1/h = "finite value lies between"  –1 and 1]`

and f(0) = k

Now, from equation (i), LHL= f(0)

`\implies` 0 = k

Hence, k = 0