lim _{x rightarrow 0} frac{x^2 log (cos x)}{l | vedclass

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(A) We are given the limit $L = \lim _{x \rightarrow 0} \frac{x^2 \log (\cos x)}{\log (1+x^2)}$.This is a $\frac{0}{0}$ indeterminate form.We use the

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(A) We are given the limit $L = \lim _{x \rightarrow 0} \frac{x^2 \log (\cos x)}{\log (1+x^2)}$.This is a $\frac{0}{0}$ indeterminate form.We use the standard limits $\lim _{u \rightarrow 0} \frac{\log (1+u)}{u} = 1$ and the Taylor series expansion for $\log(\cos x)$.First,$\log(1+x^2) \approx x^2$ as $x \rightarrow 0$.Next,$\cos x \approx 1 - \frac{x^2}{2}$,so $\log(\cos x) \approx \log(1 - \frac{x^2}{2}) \approx -\frac{x^2}{2}$.Substituting these into the limit:$L = \lim _{x \rightarrow 0} \frac{x^2 \cdot (-\frac{x^2}{2})}{x^2} = \lim _{x \rightarrow 0} (-\frac{x^2}{2}) = 0$.

$\lim _{x \rightarrow 0} \frac{x^2 \log (\cos x)}{\log (1+x^2)} = $

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