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I really thank you for your answers to my first question--I could easily solve first problem and a few more ones without another question.

But a while later I got another one while studying, then I attempted to solve it on my own. Despite my efforts to solve I have spent scores of minutes with neither finding one correct solution nor making any process.

Here is the problem:

$$ \lim_{x \rightarrow 0} \frac{e^{1-\sin x}-e^{1-\tan x}}{\tan x-\sin x} $$

The happiness would be mine, if you could let me know how to solve or even a few hints.


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3 Answers 3

up vote 5 down vote accepted

In this case you can use the mean value theorem to see that there is a $y$ in the interval $[1-\tan x,1-\sin x]$ such that

$$\frac{e^{1-\sin x}-e^{1-\tan x}}{\tan x -\sin x}=e^y$$ so the expression limits to $e^1=e$

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The expression limits to $e$, not $1$. –  nbubis Jul 6 at 0:05
oh sorry, thanks. –  Rene Schipperus Jul 6 at 0:06
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Use the expansion $\exp(1-x) \approx e - ex + O(x^2)$

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$$\frac{e^{1-\sin x}-e^{1-\tan x}}{\tan x-\sin x} $$

$$=e\cdot\frac{e^{(\tan x-\sin x)}-1}{\tan x-\sin x}\cdot\frac1{e^{\tan x}}$$

Use $\displaystyle\lim_{h\to0}\frac{e^{(h)}-1}h=1$

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@hjjg200, How about this –  lab bhattacharjee Jul 6 at 14:15
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