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The values of $p$ and $q$ for which the function $f(x)=\left\{\begin{array}{1 1}\large\frac{\sin(p+1)x+\sin x}{x} & x < 0 \\ q & x=0 \\ \large\frac{\sqrt{x+x^2}-\sqrt x}{x^3/2} & x > 0 \end{array}\right.$ is continuous for all $x\;in\;R$ are

$\begin{array}{1 1}(a)\;p=5/2,q=1/2&(b)\;p=-3/2,q=1/2\\(c)\;p=1/2,q=3/2&(d)\;p=1/2,q=-3/2\end{array}$

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LHL=$\lim\limits_{x\to 0}f(x)=\lim\limits_{h\to 0}\large\frac{\sin[(p+1)(-h)]-\sin(-h)}{-h}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{-\sin(p+1)h}{-h}+\frac{\sin(-h)}{-h}$
$\Rightarrow p+1+1=p+2$
RHL=$\lim\limits_{x\to 0}f(x)=\lim\limits_{h\to 0}\large\frac{\sqrt{1+h}-1}{h}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{1}{(\sqrt{1+h}+1)}=\frac{1}{2}$
$f(0)=q$
$\Rightarrow p=-\large\frac{3}{2}$$,q=\large\frac{1}{2}$
Hence (b) is the correct answer.
answered Dec 31, 2013 by sreemathi.v
 

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