# Limit $\lim_{x\to+\infty} {e^{2x}−1\over e^x−1}$

I am having problem in solving this indetermination:

$$\lim_{x\to+\infty} {e^{2x}−1\over e^x−1}$$

I tried to leave the term in common in evidence and cut them. I also tried to separate the limit for other notable limits but I always end up with the wrong solution.

Could you guys give a hint, please? Thank you

• Are you allowed to use de l'Hôpital formula? Feb 27, 2017 at 14:43
• I don't know, my teacher never told me about that. I am a senior in high school from Portugal.
– GiuR
Feb 27, 2017 at 14:44
• Are you able to handle $\lim_{t\to\infty}(t^2-1)/(t-1)$ ?
– user65203
Feb 28, 2017 at 9:11

Hint: Let $u=e^x$. We then have

$$\lim_{x\to+\infty} {e^{2x}−1\over e^x−1}=\lim_{u\to+\infty} {u^2−1\over u−1}=\lim_{u\to\infty}\frac{(u+1)(u-1)}{u-1}$$

• There's no need to do a change of variables, just factor the numerator directly. This just makes it harder since now you have to worry about changing the variable under the limit too. That's unnecessary. Feb 28, 2017 at 2:21
• @Mehrdad It may be unnecessary, especially to you and I, but to many like the OP, the step only makes it more clear. Feb 28, 2017 at 2:25
• @Mehrdad: Change of variable or not is not that important, yet it may help to better see what's going on. The main aspect here is to detect that $e^x-1$ cancels out. (+1) Feb 28, 2017 at 8:29

Perhaps this is even simpler $$\lim_{x\to+\infty} {e^{2x}−1\over e^x−1}=\lim_{x\to+\infty} {e^{2x}−1\over e^x−1}\left({e^{-x}\over e^{-x}}\right)=\lim_{x\to+\infty} {e^{x}−e^{-x}\over 1−e^{-x}}={\infty - 0 \over 1-0}=\infty$$

observe that for $x>0$ we have $$\frac{e^{2x}-1}{e^x-1}>e^x$$ so the answer is $\infty$

• ... also for $x<0$ Feb 28, 2017 at 7:10

You could also note that because $e^{x}e^{x} = e^{x+x} = e^{2x}$, that $$e^{2x}-1 = (e^{x}-1)(e^{x}+1),$$ then directly see that you are seeking $$\lim_{x \to +\infty}e^{x}+1.$$

This is VERY similar to the answer by Simply Beautiful Art, but doesn't require us to know anything about the legitimacy of his type of substitution in limits. Although, this probably is just as good a time as any to learn when you can use the type of substitution that was suggested, for example, see the discussion at Formal basis for variable substitution in limits

No need to be nifty enough to notice that the fraction factors. Just at a glance, looking for dominant terms, we see immediately that the numerator behaves like $e^{2x}$, the denominator like $e^{x}$.

So as a fraction, it behaves like $\frac{e^{2x}}{e^{x}}=e^{x}$, i.e. this fraction in the limit will approach $e^{x}$ so its limit is also $\infty$.

IMO trying to factor (which really only works in specialized problems) is a distraction from learning the underlying intuition, which will serve you much more generally.

You should also try and get comfortable thinking graphically; here's the graph to confirm the above.

$\frac{e^{2x} - 1}{e^x - 1} \gt \frac{e^{2x} - 1}{e^x }$
for large enough x, since we are increasing the denominator.

Rewrite as $\frac{e^{2x} - 1}{e^x - 1} \gt {e^x - e^{-x}}$ which limits to infinity.

You can use L'Hôpital's rule to find the solution. This rule says you can differentiate both numerator and denominator to find the limit: