Lim x - 0ex-1-1-cos x

Lim_x→ 0e^x 1/√(1 cosx) =lim_x→ 0e^x 1/√(1 cosx)×√(1+cosx)/√(1+cosx) =lim_x→ 0(e^x 1)√(1+cosx)/√(1 cos^2x) =lim_x→ 0(e^x 1)√(1+cosx)/sinx =lim_x→ 0 ((e^x 1)/x ) ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 0ex-1...

Question

${lim}_{x \to 0} \frac{e^{x} - 1}{\sqrt{1 - c o s x}}$

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Solution

${lim}_{x \to 0} \frac{e^{x} - 1}{\sqrt{1 - c o s x}}$

$= {lim}_{x \to 0} \frac{e^{x} - 1}{\sqrt{1 - c o s x}} \times \frac{\sqrt{1 + c o s x}}{\sqrt{1 + c o s x}}$

$= {lim}_{x \to 0} \frac{(e^{x} - 1) \sqrt{1 + c o s x}}{\sqrt{1 - c o s^{2} x}}$

$= {lim}_{x \to 0} \frac{(e^{x} - 1) \sqrt{1 + c o s x}}{s i n x}$

$= {lim}_{x \to 0} (\frac{(e^{x} - 1)}{x}) \frac{1}{(\frac{s i n x}{x})} (\sqrt{1 + c o s x})$

$= {lim}_{x \to 0} \frac{e^{x} - 1}{x} \frac{1}{{lim}_{x \to 0} (\frac{s i n x}{x})} {lim}_{x \to 0} (\sqrt{1 + c o s x})$

$= 1 \times \frac{1}{1} \times \sqrt{1 + 1}$

$= \sqrt{2}$

Suggest Corrections

limx→0ex−1√1−cosx

limx→0ex−1√1−cosx =limx→0ex−1√1−cosx×√1+cosx√1+cosx =limx→0(ex−1)√1+cosx√1−cos2x =limx→0(ex−1)√1+cosxsinx =limx→0((ex−1)x)1(sinxx)(√1+cosx) =limx→0ex−1x1limx→0(sinxx)limx→0(√1+cosx) =1×11×√1+1 =√2

${lim}_{x \to 0} \frac{e^{x} - 1}{\sqrt{1 - c o s x}}$