Question 361546
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow 0}\ \frac{x\cos(x)\ -\ \sin(x)}{x}]


This is the indeterminate form *[tex \LARGE \frac{0}{0}]


so apply L'Hôpital's Rule:



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow c}\ \frac{f(x)}{g(x)}\ =\ \lim_{x\rightarrow c}\ \frac{f'(x)}{g'(x)}]


Using the Product and Sum rules on the numerator:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow 0}\ \frac{cos(x)\ -\ x\sin(x)\ -\ \cos(x)}{1}\ =\ \lim_{x\rightarrow 0}\ \frac{x\sin(x)}{1}\ =\ 0]


b.  Works the same way.


c.  You need to combine the two fractions so that you will have a *[tex \LARGE \frac{0}{0}] indeterminate form.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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