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Lim X → 0 X ( E X − 1 ) 1 − Cos X - Mathematics

\[\lim_{x \to 0} \frac{x\left( e^x - 1 \right)}{1 - \cos x}\]

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Solution

\[\lim_{x \to 0} \left[ \frac{x \left( e^x - 1 \right)}{1 - \cos x} \right]\]

Dividing the numerator and the denominator by x2:

\[= \lim_{x \to 0} \left[ \frac{\left( \frac{e^x - 1}{x} \right)}{\left( \frac{1 - \cos x}{x^2} \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{e^x - 1}{x} \times \frac{1}{\frac{2 \sin^2 \left( \frac{x}{2} \right)}{4 \times \frac{x^2}{4}}} \right]\]
\[ = 1 \times \frac{2}{1^2}\]
\[ = 2\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 42 | Page 72
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