Question

The value of p and q for which the function

f(x) = `{((sin ("p" + 1)x + sin x)/x","  x < 0),(q ","  x = 0),((sqrt(x + x^2) - sqrtx)/(x^(3//2))","  x > 0):}`

is continuous for all x in R, are

Options

• p = `1/2`, q = `- 3/2`

• p = `5/2`, q = `1/2`

• p = `-3/2`, q = `1/2`

• p = `1/2`, q = `3/2`

Answer 1

p = `-3/2`, q = `1/2`

Explanation:

Since, f(x) is continuous for all x in R.

∴ f(x) is continuous al x = 0.

∴ f(0) = `lim_(x->0^-)` f(x)

`=> "q" = lim_(x->0) (sin (p + 1)x + sin x)/x`

`=> "q" = lim_(x->0) [("p + 1") xx (sin ("p + 1")x)/(("p + 1")x) + (sin x)/x]`

⇒ q = (p + 1) + 1

⇒ q = p + 2

The values of p and q in option (C) satisfies this condition.