The value of $p$ and $q$ for which the function $$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} \cr } } \right.$$ is continuous for all $x$ in R, are

Question

The value of $p$ and $q$ for which the function

$$f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} \cr } } \right.$$

is continuous for all $x$ in R, are

$p =$ ${5 \over 2}$, $q = $ ${1 \over 2}$

$p =$ $-{3 \over 2}$, $q = $ ${1 \over 2}$

$p =$ ${1 \over 2}$, $q = $ ${3 \over 2}$

$p =$ ${1 \over 2}$, $q = $ $-{3 \over 2}$

Solution

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