ₓarrow0}≤ft( 2x/2x)^{{1}{x²}} =

Correct Answer is : e23e²/3}e32

Correct Answer is : e23e²/3}e32

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TS EAMCET 04 May 2019 Shift 1 Solved Paper

Section: Mathematics

Question : 62 of 160

Marks: +1, -0

\lim\limits_{x\rightarrow0}\left(\frac{\sinh 2x}{2x}\right)^{\frac{1}{x^{2}}}

Solution:

Consider the expression,

\lim\limits_{x\rightarrow0}\left(\frac{\sinh 2x}{2x}\right)^{\frac{1}{x^{2}}}

Simplify the above

\lim\limits_{x\rightarrow0}\left(\frac{e^{2x}-e^{-2x}}{4x}\right)^{\frac{1}{x^{2}}}\quad=\lim\limits_{x\rightarrow0}\left(\frac{e^{4x}-1}{4e^{2x}x}\right)^{\frac{1}{x^{2}}}

\quad=\lim\limits_{e^{x}\rightarrow0}\left(\frac{e^{4x}-1}{4e^{2x}x}\right)\frac{1}{x^{2}}

\quad=\lim\limits_{x^{x}\rightarrow0}\left(\frac{e^{4x}-1-4xe^{2x}}{4x^{3}e^{2x}}\right)

\quad=\lim\limits_{e^{x}\rightarrow0}\left(\frac{4e^{4x}-0-4e^{2x}-8xe^{2x}}{4(3x^{2}e^{2x}-2x^{3}e^{2x})}\right)

Apply the L'Hospital rule,

\lim\limits_{e^{x}\rightarrow0}\left(\frac{e^{2x}-1-2x}{3x^{2}-2x^{3}}\right)

=\lim\limits_{e^{x}\rightarrow0}\left(\frac{2e^{2x}-2}{6x-x^{2}}\right)

\quad=\lim\limits_{e^{x}\rightarrow0}\left(\frac{e^{2x}-1}{3x(1-x^{2})}\right)

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

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