If f (x) = {cases} (1+| x|)^{a/| x|}, & -π/6 , then the value of a and b, if f is continuous at x = 0, are respectively

Correct Answer is : 232/332 , e23e²/3}e32

Correct Answer is : 232/332 , e23e²/3}e32

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VITEEE 2013 Paper

Section: Mathematics

Question : 117 of 120

Marks: +1, -0

If f (x) =

\begin{cases} (1+|\sin x|)^{\frac{a}{|\sin x|}}, & -\pi/6 < x < 0 \\ e^{\frac{\tan 2x}{\tan 3x}}, & 0 < x < -\pi/6 \end{cases}

Solution:

For f (x) to be continuous at x = 0

\lim\limits_{x\to 0^{-}}

f (x) = f (0) =

\lim\limits_{x\to 0^{+}}

f (x)

\lim\limits_{x\to 0^{-}}

(1+|\sin x|)^{\frac{a}{|\sin x|}}

e^{\lim\limits_{x\to 0}\left\{\frac{|\sin x| a}{|\sin x|}\right\}}

e^{a}

Now,

\lim\limits_{x\to 0^{+}}

e^{\frac{\tan 2x}{\tan 3x}}

\lim\limits_{x\to 0^{+}}

e^{\frac{\left(\frac{\tan 2x}{2x}\cdot 2x\right)}{\left(\frac{\tan 3x}{3x}\cdot 3x\right)}}

\lim\limits_{x\to 0^{+}}

e^{\frac{2}{3}}

e^{\frac{2}{3}}

Since, f(x) is continuous at x = 0

∴

e^{a}

e^{\frac{2}{3}}

⇒ a =

\frac{2}{3}

and b =

e^{\frac{2}{3}}

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