If f(x) = begincases ( sin (p +1) + sin x/x) &, x 0 endcases is continuous at x = 0, then the ordered pair (p,q) is equal to :

Question

If f(x) = begincases ( sin (p +1) + sin x/x) &, x < 0 q &, x = 0 (√x+x2 - √x/x3/2) &, x > 0 endcases is continuous at x = 0, then the ordered pair (p,q) is equal to : (A) ((5/2) , (1/2)) (B) ( - (3/2) , - (1/2)) (C) ( - (1/2) , (3/2)) (D) ( - (3/2) , (1/2))

Tardigrade Admin · Accepted Answer

RHL = limx→0+ (√x+x2 -√x/x(3)2) = limx→0+ (√1+x-1/x) = (1/2) LHL = limx→0 ( sin(p+1)x+ sin x/x) =(p+1)+1 = p+2 for continuity LHL = RHL = f(0) ⇒ (p,q) = ((-3/2) , (1/2) )

Q. If $f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) + s i n x}{x} q \frac{x + x ^{2} - x}{x ^{3/2}}, x < 0, x = 0, x > 0$
is continuous at $x = 0$ , then the ordered pair $(p, q)$ is equal to :

$(\frac{5}{2}, \frac{1}{2})$

$(- \frac{3}{2}, - \frac{1}{2})$

$(- \frac{1}{2}, \frac{3}{2})$

$(- \frac{3}{2}, \frac{1}{2})$

$R H L = lim_{x \to 0^{+}} \frac{x + x ^{2} - x}{x ^{\frac{3}{2}}} = lim_{x \to 0^{+}} \frac{1 + x - 1}{x} = \frac{1}{2}$
$L H L = lim_{x \to 0} \frac{s i n ( p + 1 ) x + s i n x}{x} = (p + 1) + 1 = p + 2$ for continuity $L H L = R H L = f (0)$
$\Rightarrow (p, q) = (\frac{- 3}{2}, \frac{1}{2})$

Q. If f(x)=⎩⎨⎧​xsin(p+1)+sinx​qx3/2x+x2​−x​​​,x<0,x=0,x>0​ is continuous at x=0, then the ordered pair (p,q) is equal to :

(25​,21​)

(−23​,−21​)

(−21​,23​)

(−23​,21​)