Lim x - 0ex-esin x-x-sin x

Lim_x → 0e^x e^sin x /x sin x = lim_x → 0[e^sin x [e^x/e sin x 1 ]/ x sin x ] as x →∞, x sin x → 0 = lim_x → 0 e^sin x [e^x sin x 1/x si ...

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Standard Limits to Remove Indeterminate Form

lim x → 0ex-e...

Question

${lim}_{x \to 0} \frac{e^{x} - e^{sin x}}{x - sin x}$

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Solution

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lim x \to 0 (\frac{e^{x} - e^{sin x}}{x - sin x}) =

lim x \to 0 \frac{e^{x} - e^{sinx}}{x - sin x} .

lim x \to 0 \frac{e^{x} - e^{sin x}}{x - sin x} = 1

${lim}_{x \to 0} \frac{e^{x} - 1 + s i n x}{x}$

lim x \to 0 \frac{e^{x} + sin x - 1}{log (1 + x)}

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