L'Hospital's rule:Example Needed

Question

I just thought of a certain question regarding L'Hospital's rule.

The rule can be applied in indeterminate functions of form $f(x)/g(x)$

Are there any example where f'(x)/g'(x) is again indeterminate and f''(x)/g''(x) and so on indefinitely (like its looping)?What should be the approach in those particular cases?

I'm just going through Calc 2 course..so please suit your answer to my level of understanding.

The classical example is $f(x)=x$ and $g(x)=\sqrt{1+x^2}$; after rearranging $f'(x)/g'(x)$ you get $\frac{\sqrt{1+x^2}}{x}$ and the next application of l'Hôpital brings you at the starting point. — egreg, Aug 09 '15 at 17:13
see here http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx — Dr. Sonnhard Graubner, Aug 09 '15 at 17:14
A joke, but in principle correct: Find $\lim_{x\to\infty} \frac{e^x}{e^x}$ using the Rule. — André Nicolas, Aug 09 '15 at 17:16

Aleksandar · Answer 1 · 2015-08-09T17:17:20.773

2

Yes a good example is $f(x)=e^{ax}$ and $g(x)=e^{bx}$. $a,b \neq 0$

$$\lim_{x \to \infty} \frac{e^{ax}}{e^{bx}}=\lim_{x \to \infty} \frac{a^n e^{ax}}{b^n e^{bx}}$$

In short the ratio between two exponential functions as $x$ goes to $\infty$ cannot be solved by L'Hopitals Rule.

edited Aug 09 '15 at 17:17

answered Aug 09 '15 at 17:14

Aleksandar

1,771

Assumed that $a,b$ are not zero and have the same sign. – ajotatxe Aug 09 '15 at 17:16
@ajotatxe Yes, I was assuming that. – Aleksandar Aug 09 '15 at 17:16
so how should it be solved? – Aug 09 '15 at 17:16
1

$\frac{e^{ax}}{e^{bx}}=e^{(a-b)x}$. – ajotatxe Aug 09 '15 at 17:18
It is $\infty$, $\lim e^{x(a-b)}=\infty$ – Aleksandar Aug 09 '15 at 17:18
In this example: 1) $b < 0$ or 2) $a > b$ – Leucippus Aug 09 '15 at 17:24

score 1 · Answer 2 · answered Aug 09 '15 at 17:14

1

Make for example $f(x)=2^x$ and $g(x)=3^x$ and the limit at $x\to\infty$.

Of course, the limit can be easily calculated by other means.

answered Aug 09 '15 at 17:14

ajotatxe

65,084

Such as $\lim (\frac{2}{3})^x$ – Aleksandar Aug 09 '15 at 18:32

L'Hospital's rule:Example Needed

2 Answers2