If - fx- begin cases - gx-011ddfrac - isqrt x-x- 2 -lsqrt -xx-3 - 2- x gt 0vend cases iscontinuousat x-0- thentheorderedpairp-q- is equal to -

For f(x) to be continuous at x=0 , f(0)=f(0^+)=f(0^ ) f(0)=q f(0^+)=lim_x→ 0^+√(x+x^2) √(x)x^3/2 =lim_x→ 0^+√(1+x) √(1)x Applying L hospital's rule nbsp; =1 ...

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Standard XII

Mathematics

Continuity of a Function

If $ fx= begi...

Question

If $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (p + 1) x + sin x}{x} & , x < 0 q & x = 0 \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{matrix}$ is continuous at $x = 0$ , then the ordered pair $(p, q)$ is equal to :

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

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

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

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

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Solution

The correct option is A $(- \frac{3}{2}, \frac{1}{2})$
For $f (x)$ to be continuous at $x = 0$ , $f (0) = f (0^{+}) = f (0^{-})$
$f (0) = q . . . (1) f (0^{+}) = lim x \to 0^{+} \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} = lim x \to 0^{+} \frac{\sqrt{1 + x} - \sqrt{1}}{x}$

Applying L' hospital rule
$= \frac{1}{2}$
$f (0^{-}) = lim x \to 0^{-} \frac{sin (p + 1) x + sin x}{x}$
Applying L' hospital rule
$= lim x \to 0^{-} \frac{cos (p + 1) x \cdot (1 + p) + cos x}{1}$ $= 2 + p \Rightarrow p + 2 = q = \frac{1}{2} \Rightarrow p = \frac{- 3}{2}, q = \frac{1}{2}$

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If f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩sin(p+1)x+sinxx,x<0 qx=0√x+x2−√xx3/2,x>0 is continuous at x=0, then the ordered pair (p,q) is equal to :

If $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (p + 1) x + sin x}{x} & , x < 0 q & x = 0 \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{matrix}$ is continuous at $x = 0$ , then the ordered pair $(p, q)$ is equal to :