Determine the following limits $\lim_{x \to 0}\left(\sin x + \frac{e^x-1}{x}\right)$ and $\lim_{x \to 0}\frac{e^x-1}{\sin x}$

Question

Determine the following limits

$\lim_{x \to 0}\left(\sin x + \frac{e^x-1}{x}\right)$
$\lim_{x \to 0}\frac{e^x-1}{\sin x}$

For 1, see here. For 2, see here. (Found using Approach0.xyz) — Workaholic, Dec 02 '16 at 18:23
Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be put on hold, see meta. — Martin Sleziak, Dec 02 '16 at 19:03

score 2 · Answer 1 · answered Dec 02 '16 at 17:34

2

For example, using the definition of derivative:

$$\lim_{x\to0}\frac{e^x-1}{\sin x}=\lim_{x\to0}\frac{e^x-1}x\cdot\frac x{\sin x}=e'(0)\cdot1=1$$

Can you now do the other one?

answered Dec 02 '16 at 17:34

DonAntonio

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score 2 · Accepted Answer · answered Dec 02 '16 at 17:35

2

we have $$\lim_{x\to 0}\left(\sin(x)+\frac{e^x-1}{x}\right)=0+\lim_{x \to 0}e^x=1$$ further we get $$\lim_{x \to 0}\left(\frac{e^x-1}{\sin(x)}\right)=\lim_{x \to 0}\frac{e^x}{\cos(x)}=1$$

answered Dec 02 '16 at 17:35

Dr. Sonnhard Graubner

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Funny: the OP tagged his question "limits-without-l'Hospital", yet both these limits are calculated via l'Hospital in this answer...and it gets accepted... – DonAntonio Dec 02 '16 at 19:04

score 1 · Answer 3 · answered Dec 02 '16 at 17:33

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Hint. Recall that $e^x=1+x+o_{x\to 0}(x)$ and $\sin(x)=x+o_{x\to 0}(x)$. These asymptotics are derived from Taylor's theorem.

answered Dec 02 '16 at 17:33

C. Falcon

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Determine the following limits $\lim_{x \to 0}\left(\sin x + \frac{e^x-1}{x}\right)$ and $\lim_{x \to 0}\frac{e^x-1}{\sin x}$

3 Answers3