The value of p and q for which the function

The value of p and q for which the function

f(x) = (sin(p + 1)x + sinx)/x, x < 0
f(x) = q, x = 0
f(x) = (√(x + x2) - √x)/x3/2, x > 0

is continuous for all x in R, are:

  1. p = 1/2 , q = 3/2
  2. p = -3/2 , q = 1/2
  3. p = 1/2 , q = -3/2
  4. p = 5/2 , q = 1/2

Answer

f(0) = q

f(0+) = limx→0+ ((1 + x1/2)-1)/x = limx→0+ 1 + 1/2x + ... -1/x = 1/2

f(0-) = limx→0- (sin(p + 1)x + sinx)/x 

f(0-) = limx→0- (cos(p + 1)x)(p + 1) + (cosx)

= (p + 1) + 1

The correct option is B.