Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How do you solve this limit? I know this is probably really easy.

$$ \lim_{x \to ∞} \left(f(x) = (1 / x) * e ^ x\right) $$

share|improve this question
1  
Have you learnt L'hopital's rule? –  Oria Gruber Jul 7 at 14:43
    
@OriaGruber No, I am teaching myself mathematics. Thanks, didn't know about this. –  Pacha Jul 7 at 14:45
    
Alternately, how are you feeling about Taylor series? –  Jason Knapp Jul 7 at 14:45
    
Sandwich also works great here. –  Oria Gruber Jul 7 at 14:50

5 Answers 5

up vote 5 down vote accepted

Try L'Hospital's Rule: $$ \lim_{x\to \infty}\frac{f(x)}{g(x)} = \lim_{x\to \infty}\frac{f'(x)}{g'(x)}. $$

share|improve this answer

Another way would be the sandwich method:

Notice that $e^{0.5x}$ is monotonic ascending, and et's all agree that after a certain point $n$, $e^{0.5x} >x$ for all $x > n$ (Showing this is pretty easy and I will leave it to you).

And so, if $x>n$:

$$\frac{e^x}{e^{0.5x}} < \frac{e^x}{x} < e^x $$

When $x$ tends to infinity. the left side limit approaches infinity, the right side limit approaches to infinity, so what's bound in the middle must approach infinity as well.

share|improve this answer

For all $x\in\mathbb{R}$, we have $e^{x/2}\ge1+x/2$. Therefore, for $x\gt0$, $e^x\ge\left(1+x/2\right)^2$, and so $$ \begin{align} \lim_{x\to\infty}\frac1xe^x &\ge\lim_{x\to\infty}\frac1x\left(1+x/2\right)^2\\ &\ge\lim_{x\to\infty}x/4\\ &\to\infty \end{align} $$

share|improve this answer

Note also that by L'Hospital's Rule, $e^x$ grows quicker than any polynomial as $x$ approaches $\infty$.

share|improve this answer
1  
In fact, for $x\ge0$, we can modify the idea in my answer to get $$e^x\ge\left(1+\frac xn\right)^n$$ –  robjohn Jul 8 at 9:52

The L'Hospital rule is applicable here, however the other answers fail to stress its limitations. Be sure to read the wikipedia article.

share|improve this answer
1  
In my humble opinion, it would be more appropriate to post this as a comment. Might I suggest you read the about page –  gebruiker Jul 7 at 18:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.