The values of p and q for which the function f x - sin p-1 x-sin x - x x0 -

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Standard X

Mathematics

Distance Formula

The values of...

Question

The values of $p$ and $q$ for which the function
$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (p + 1) x + sin x}{x} x < 0 q x = 0 \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{\frac{3}{2}}} x > 0 \end{matrix}$

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Solution

$\begin{matrix} f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (p + 1) x + sin x}{x} x < 0 q x = 0 \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{\frac{3}{2}}} x > 0 \end{matrix} H e r e, f (0) = q f (0^{+}) = {lim}_{x \to 0^{+}} \frac{{(1 + x)}^{\frac{1}{2}} - 1}{x} = {lim}_{x \to 0^{+}} \frac{1 + \frac{x}{2} + . . . . . + 1}{x} = \frac{1}{2} f (0^{-}) = {lim}_{x \to 0^{-}} \frac{sin (p + 1) x + sin x}{x} = {lim}_{x \to 0^{-}} \frac{cos (p + 1) x . (p + 1) + cos x}{1} = p + 1 + 1 = p + 2 p + 2 = \frac{1}{2} p = \frac{- 3}{2}, q = \frac{1}{2} \end{matrix}$

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Similar questions

Q. Determine the values of

a, b, c

for which

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} f (x) = \frac{sin (a + 1) x + sin x}{x} & for x < 0 = c & for x = 0 \frac{(x + b x^{2})^{\frac{1}{2}} - x^{\frac{1}{2}}}{b x^{\frac{1}{2}}} & for x > 0 \end{matrix}

is continuous at

x = 0

Q. The values of

a, b

and

c

which make the function

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (a + 1) x + sin x}{x} & , x < 0 c & , x = 0 \frac{\sqrt{x + b x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{matrix}

continuous at

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are

Find the value of
$a$ for which the function f defined by

$f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} a s i n \frac{π}{2} (x + 1) & , x \leq 0 \frac{t a n x - s i n x}{x^{3}} & , x > 0 \end{matrix}$ is
continuous at $x = 0$

Q. Find the value of 'a' for which the function f defined as

f (x) = ⎧ ⎨ ⎩ \begin{matrix} a sin \frac{π}{2} (x + 1), x \leq 0 \frac{tan x - sin x}{x^{3}}, x > 0 \end{matrix} ⎫ ⎬ ⎭

is continuous at

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f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{(x + b x^{2})^{1 /_{2}} - x^{1 / 2}}{b x^{3 / 2}} & x > 0 c & x = 0 \frac{sin (a + 1) x + sin x}{x} & x < 0 \end{matrix}

is continuous at

x = 0

, then

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The values of p and q for which the functionf(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩sin(p+1)x+sinxxx<0qx=0√x+x2−√xx32x>0

f(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩sin(p+1)x+sinxxx<0qx=0√x+x2−√xx32x>0Here,f(0)=qf(0+)=limx→0+(1+x)12−1x=limx→0+1+x2+.....+1x=12f(0−)=limx→0−sin(p+1)x+sinxx=limx→0−cos(p+1)x.(p+1)+cosx1=p+1+1=p+2p+2=12p=−32,q=12

The values of $p$ and $q$ for which the function
$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (p + 1) x + sin x}{x} x < 0 q x = 0 \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{\frac{3}{2}}} x > 0 \end{matrix}$