The value of lim_{x to 0} (cos ax)^{csc^2 bx} | vedclass

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(B) Let $L = \lim_{x 	o 0} (\cos ax)^{\csc^2 bx}$. This is of the form $1^\infty$.Using the formula $\lim_{x 	o 0} f(x)^{g(x)} = e^{\lim_{x 	o 0} g

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$e^{-\frac{a^2}{2b^2}}$

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(B) Let $L = \lim_{x 	o 0} (\cos ax)^{\csc^2 bx}$. This is of the form $1^\infty$.Using the formula $\lim_{x 	o 0} f(x)^{g(x)} = e^{\lim_{x 	o 0} g(x)(f(x)-1)}$,we have:$L = e^{\lim_{x 	o 0} \csc^2 bx (\cos ax - 1)}$$L = e^{-\lim_{x 	o 0} \frac{1 - \cos ax}{\sin^2 bx}}$Using the identity $1 - \cos 	heta = 2\sin^2(	heta/2)$ and $\lim_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$:$L = e^{-\lim_{x 	o 0} \frac{2\sin^2(ax/2)}{\sin^2 bx}} = e^{-\lim_{x 	o 0} \frac{2(ax/2)^2}{(bx)^2}} = e^{-\frac{2(a^2/4)}{b^2}} = e^{-\frac{a^2}{2b^2}}$

The value of $\lim_{x \to 0} (\cos ax)^{\csc^2 bx}$ is

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