1
$\begingroup$

Evaluate $\lim_{s\to0}\frac{s\sqrt{e^s}}{e^s-1}$ without using L'Hospital rule.

I've done this: Since $\sqrt{e^s}\to1$as $s\to1$, so $\lim_{s\to0}\frac{s\sqrt{e^s}}{e^s-1}=\lim_{s\to0}\frac{s}{e^s-1}=1$

Are my steps correct? Is step 1 to step 2 available? Thank you.

  • 0
    What's your justification for the last limit being 1 without using L'Hopital?2012-08-21
  • 0
    @KevinCarlson: Usually, it follows straightforward from $(1+x)^\frac1x \to \mathrm e$ with $x\to 0$. **To Jason:** yes, since limits of both terms do exist, the limit of the product exists and equals to the product of limits.2012-08-21
  • 1
    @Ilya $(1+x)^{1/x} \to e$ as $x \to 0$!2012-08-21
  • 0
    @Mercy: sorry, that was a typo - I was already thinking of equivalent pairs like $\log(1+x)\sim x$.2012-08-21
  • 1
    Hint for an alternative derivation (yours is right as shown by Ilya) : $\displaystyle \frac{s\sqrt{e^s}}{e^s-1}=\frac{s}{e^{\frac s2}-e^{-\frac s2}}$ and since the $\sinh\cdots$2012-08-21

1 Answers 1

3

Well, indeed you are right: you can prove it just based on the limit $\lim\limits_{x\to 0}(1+x)^{1/x} = \mathrm e$. By taking the logarithm of both parts, you obtain $\lim\limits_{x\to 0}\frac{\log(1+x)}{x} = 1$. If you put $s =\log(1+x)$ then you get $$ \lim\limits_{s\to 0}\frac{s}{\mathrm e^s-1} = 1 $$ as you mentioned. Now, if you have $\lim\limits_{s\to 0}f(s) = F$ and $\lim\limits_{s\to 0}g(s) = G$ then $$ \lim\limits_{s\to 0}f(s)g(s) = FG $$ and the limit in LHS exists - that's why you can go from step 1 to step 2 in your proof.

  • 0
    @jasoncube: you are welcome2012-08-21
  • 0
    One could even make again the substitution $s= log(t+1)$ and derives again at the desired expression $\frac{\log(1+t)}{t}$2017-10-10