3^x 2^xx=2^x((32)^0+x 1)x #160; #160; #160; #160; #160; =2^x((32)^0+x (32)^x)x Now, lim_x→ 02^x((32)^0+x (32)^x)x =lim_x→ 02^ ...

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Substitution Method to Remove Indeterminate Form

Compute Lim...

Question

Compute $L i m \frac{3^{x} - 2^{x}}{x}$

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Solution

$\frac{3^{x} - 2^{x}}{x} = \frac{2^{x} ({(\frac{3}{2})}^{0 + x} - 1)}{x}$
$= \frac{2^{x} ({(\frac{3}{2})}^{0 + x} - {(\frac{3}{2})}^{x})}{x}$
Now, ${lim}_{x \to 0} \frac{2^{x} ({(\frac{3}{2})}^{0 + x} - {(\frac{3}{2})}^{x})}{x}$
$= {lim}_{x \to 0} 2^{x} \times {lim}_{x \to 0} \frac{({(\frac{3}{2})}^{0 + x} - {(\frac{3}{2})}^{x})}{x}$
$= 1 {⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{d {(\frac{3}{2})}^{x}}{d x} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}_{x = 0}$
$= {log}_{e} (\frac{3}{2})$

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