Lim_x→ 0a^x+a^ x 2/x^2=lim_x→ 0a^x+1/a^x 2/x^2 =lim_x→ 0a^2x 2a^x+1/a^2.x^2 =lim_x→ 0 ( a^x 1/x )×1/a^x =(log a)^2×1/a^0 =(log a)^2 ...

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Substitution Method to Remove Indeterminate Form

lim x → 0ax+a...

Question

${lim}_{x \to 0} \frac{a^{x} + a^{- x} - 2}{x^{2}}$

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Solution

${lim}_{x \to 0} \frac{a^{x} + a^{- x} - 2}{x^{2}} = {lim}_{x \to 0} \frac{a^{x} + \frac{1}{a^{x}} - 2}{x^{2}}$

$= {lim}_{x \to 0} \frac{a^{2 x} - 2 a^{x} + 1}{a^{2} . x^{2}}$

$= {lim}_{x \to 0} (\frac{a^{x} - 1}{x}) \times \frac{1}{a^{x}}$

$= (l o g a)^{2} \times \frac{1}{a^{0}}$

$= (l o g a)^{2}$

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