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Continuity of a Function

Solve ltx →1...

Question

Solve
$l t x \to \frac{1}{2} \frac{2^{x} + 2^{3 - x} - 6}{\sqrt{2^{- x}} - 2^{1 - x}}$

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Solution

(say) $l = {lim}_{x \to \frac{1}{2}} \frac{2^{x} + 2^{3 - x} - 6}{\sqrt{2^{- x}} - 2^{1 - x}} = \frac{2^{1 / 2} + 2^{3 - 1 / 2} - 6}{\sqrt{2^{- 1 / 2}} - 2^{1 - 1 / 2}}$
$l = \frac{2^{1 / 2} + 2^{5 / 2} - 6}{\frac{1}{2^{1 / 4}} - 2^{1 / 2}}$
$l = \frac{(2^{1 / 2} + 2^{5 / 2} - 6) (2^{1 / 4})}{1 - 2^{3 / 4}}$
$l = 2^{3 / 4} \frac{(1 + 2^{2} - 3 \sqrt{2})}{1 - 2^{3 / 4}}$
$l = 2^{3 / 4} \frac{(5 - 3. 2^{1 / 2})}{1 - 2^{3 / 4}}$

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