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  4. The value of lim limitsx→2 (2x +23-x-6/√2-x -21-x) is

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Q. The value of $\lim\limits_{x\to2} \frac{2^{x} +2^{3-x}-6}{\sqrt{2^{-x}} -2^{1-x}}$ is

Limits and Derivatives

Solution:

$\lim\limits_{x\to2} \frac{2^{x} +2^{3-x}-6}{\sqrt{2^{-x}} -2^{1-x}}$
$ = \lim\limits_{x\to 2} \frac{\left(2^{x}\right)^{2}-6\times2^{x} + 2^{3}}{\sqrt{2^{x}}-2} $[Multiplying Nr and Dr by $2^x$]
$=\lim\limits_{x\to2} \frac{\left(2^{x}-4\right)\left(2^{x}-2\right)\left(\sqrt{2^{x}}+2\right)}{\left(\sqrt{2^{x}}-2\right)\left(\sqrt{2^{x}}+2\right)}$
$=\lim\limits_{x\to2} \frac{\left(2^{x}-4\right)\left(2^{x}-2\right)\left(\sqrt{2^{x}}+2\right)}{\left(2^{x} -4 \right)} $
$=\lim\limits_{x\to 2} \left(2^{x} -2\right)\left(\sqrt{2^{x}}+2\right) = \left(2^{2} -2\right)\left( 2 + 2\right) = 8$