The value of $p$ and $q$ for which the function $f(x)=\left\{\begin{array}{cl}\frac{\sin (p+1) x+\sin x}{x} & , x 0\end{array}\right.$ is continuous for all $x$ in $R$, is

(B) $p=-\frac{3}{2}, q=\frac{1}{2}$

(A) $p=\frac{5}{2}, q=\frac{1}{2}$

(C) $p=\frac{1}{2}, q=\frac{3}{2}$

(D) $p=\frac{1}{2}, q=-\frac{3}{2}$

JEE Main · 2011 · 10 Apr 2011 · Offline · Continuity and Differentiability

The value of

p

and

q

for which the function

f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x} q \frac{x + x ^{2} - x}{x ^{3/2}}, x < 0, x = 0, x > 0

is continuous for all

x

R

, is

Answer: (B)

Continuity at

x = 0

needs the two one-sided limits to equal

q

. From the left,

\frac{sin ( p + 1 ) x + sin x}{x} \to (p + 1) + 1 = p + 2

. From the right, rationalising,

\frac{x + x ^{2} - x}{x ^{3/2}} = \frac{x ( 1 + x - 1 )}{x ^{3/2}} = \frac{1 + x - 1}{x} \to \frac{1}{2}

. So

q = \frac{1}{2}

and

p + 2 = \frac{1}{2}

, giving

p = - \frac{3}{2}

and

q = \frac{1}{2}

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option