lim _{x rightarrow 0} frac{(1-e^x) sin x}{x^2 | vedclass

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Question

(A) We evaluate the limit: $\lim _{x \rightarrow 0} \frac{(1-e^x) \sin x}{x^2+x^3}$ Factor the denominator: $\lim _{x \rightarrow 0} \frac{(1-e^x) \si

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(A) We evaluate the limit: $\lim _{x ightarrow 0} \frac{(1-e^x) \sin x}{x^2+x^3}$ Factor the denominator: $\lim _{x ightarrow 0} \frac{(1-e^x) \sin x}{x^2(1+x)}$ Rewrite the expression: $\lim _{x}$ ${ightarrow 0} \left( \frac{1-e^x}{x} ight) 	imes \left( \frac{\sin x}{x} ight) 	imes \left( \frac{1}{1+x} ight)$ Using standard limits $\lim _{x ightarrow 0} \frac{e^x-1}{x} = 1$ and $\lim _{x ightarrow 0} \frac{\sin x}{x} = 1$: The expression becomes $(-1) 	imes (1) 	imes \left( \frac{1}{1+0} ight) = -1 	imes 1 	imes 1 = -1$

$\lim _{x \rightarrow 0} \frac{(1-e^x) \sin x}{x^2+x^3}$ is equal to

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