Question

If the following function is continuous at x = 2 then the value of k will be ______.

f(x) = `{{:(2x + 1",", if x < 2),(                 k",", if x = 2),(3x - 1",", if x > 2):}`

Options

• 2

• 3

• 5

• – 1

Answer 1

If the following function is continuous at x = 2 then the value of k will be 5.

f(x) = `{{:(2x + 1",", if x < 2),(                 k",", if x = 2),(3x - 1",", if x > 2):}`

Explanation:

k = 5

∵ f(x) is continuous at x = 2

∴ LHL = RHL = f(2)   ...(i)

Now LHL = `lim_(h -> 0) f(2 - h)`

= `lim_(h -> 0) 2(2 - h) + 1`

= `lim_(h -> 0) 5 - 2h`

= 5 – 0

= 5

RHL = `lim_(h -> 0) f(2 + h)`

= `lim_(h -> 0) 3(2 + h) - 1`

= `lim_(h -> 0) 5 + 3h`

= 5 + 0

= 5

And f(2) = k

Then by (i),

LHL = RHL = f(2)

`\implies` k = 5