lim _{x rightarrow 1} frac{2^{2 x-2}-2^x+1}{s | vedclass

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(B) Let $L = \lim _{x \rightarrow 1} \frac{2^{2 x-2}-2^x+1}{\sin ^2(x-1)}$.Note that $2^{2x-2} - 2^x + 1$ can be written as $(2^{x-1})^2 - 2 \cdot 2^{

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$(\log 2)^2$

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(B) Let $L = \lim _{x \rightarrow 1} \frac{2^{2 x-2}-2^x+1}{\sin ^2(x-1)}$.Note that $2^{2x-2} - 2^x + 1$ can be written as $(2^{x-1})^2 - 2 \cdot 2^{x-1} + 1$ is incorrect,let's re-evaluate the numerator: $2^{2x-2} - 2^x + 1 = (2^{x-1})^2 - 2 \cdot 2^{x-1} + 1 = (2^{x-1}-1)^2$.So,$L = \lim _{x \rightarrow 1} \frac{(2^{x-1}-1)^2}{\sin ^2(x-1)}$.Divide numerator and denominator by $(x-1)^2$:$L = \lim _{x \rightarrow 1} \frac{\left(\frac{2^{x-1}-1}{x-1}\right)^2}{\left(\frac{\sin(x-1)}{x-1}\right)^2}$.Using the standard limits $\lim _{h \rightarrow 0} \frac{a^h-1}{h} = \log a$ and $\lim _{h \rightarrow 0} \frac{\sin h}{h} = 1$,where $h = x-1$:$L = \frac{(\log 2)^2}{1^2} = (\log 2)^2$.

$\lim _{x \rightarrow 1} \frac{2^{2 x-2}-2^x+1}{\sin ^2(x-1)}=$

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