Evaluate the following limit: _ x 0 8^x-4^x-2^x+1 x^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{8^\text{x}-4^\text{x}-2^\text{x}+1}{\text{x}^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\bigg[\frac{8^\text{x}-4^\text{x}-2^\text{x}+1}{\text{x}^2}\bigg]$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg[\frac{\big(2^\text{x}\big)^3-\big(2^\text{x}\big)^2-2^\text{x}+1}{\text{x}^2}\Bigg]$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg[\frac{\big(2^\text{x}\big)^2\big(2^\text{x}-1\big)-1\big(2^\text{x}-1\big)}{\text{x}^2}\Bigg]$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg[\frac{\big(2^\text{2x}-1\big)\big(2^\text{x}-1\big)}{\text{x}^2}\Bigg]$
$=\lim\limits_{\text{x}\rightarrow0}\Bigg[\frac{2\big(2^\text{2x}-1\big)}{2\text{x}}\times\Big(\frac{2^\text{x}-1}{\text{x}}\Big)\Bigg]$
$=2\text{ log }2\times\text{log }2$
$=\text{ log }(2)^2\times\text{log }2$
$=(\text{ log }4)\times(\text{log }2)$

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