Lim_x→ 0sin2x/e^x 1 =lim_x→ 0sin2x/2x× 2 lim_x→ 0x/e^x 1 =1 × 2 ×1/lim_x→ 0e^x 1/x =1 × 2 × 1 =2 ...

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Standard Limits to Remove Indeterminate Form

lim x → 0sin ...

Question

${lim}_{x \to 0} \frac{s i n 2 x}{e^{x} - 1}$

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Solution

${lim}_{x \to 0} \frac{s i n 2 x}{e^{x} - 1}$

$= {lim}_{x \to 0} \frac{s i n 2 x}{2 x} \times 2 {lim}_{x \to 0} \frac{x}{e^{x} - 1}$

$= 1 \times 2 \times \frac{1}{{lim}_{x \to 0} \frac{e^{x} - 1}{x}}$

$= 1 \times 2 \times 1$

$= 2$

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