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(C) We are given the limit: $\lim _{x \rightarrow \infty} x\left(\log \left(1+\frac{x}{2}\right)-\log \frac{x}{2}\right)$.Using the property $\log a -

Vedclass · Accepted Answer

$2$

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(C) We are given the limit: $\lim _{x \rightarrow \infty} x\left(\log \left(1+\frac{x}{2}\right)-\log \frac{x}{2}\right)$.Using the property $\log a - \log b = \log \left(\frac{a}{b}\right)$,we have:$\lim _{x \rightarrow \infty} x \log \left(\frac{1+\frac{x}{2}}{\frac{x}{2}}\right) = \lim _{x \rightarrow \infty} x \log \left(\frac{2}{x} + 1\right)$.Let $t = \frac{1}{x}$. As $x \rightarrow \infty$,$t \rightarrow 0$. The expression becomes:$\lim _{t \rightarrow 0} \frac{\log (1+2t)}{t}$.Using the standard limit $\lim _{u \rightarrow 0} \frac{\log (1+u)}{u} = 1$,we multiply and divide by $2$:$\lim _{t \rightarrow 0} 2 \cdot \frac{\log (1+2t)}{2t} = 2 \cdot 1 = 2$.

$\lim _{x \rightarrow \infty} x\left(\log \left(1+\frac{x}{2}\right)-\log \frac{x}{2}\right) = $

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