# What is l'hopital: Definition and 35 Discussions

Guillaume François Antoine, Marquis de l'Hôpital (French: [ɡijom fʁɑ̃swa ɑ̃twan maʁki də lopital]; sometimes spelled L'Hospital; 1661 – 2 February 1704), also known as Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont, and Seigneur d'Ouques-la-Chaise, was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞. Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his 1696 treatise on the infinitesimal calculus, entitled Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. This book was a first systematic exposition of differential calculus. Several editions and translations to other languages were published and it became a model for subsequent treatments of calculus.

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1. ### L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?

I'm talking about the x^(-4/3) how does it equal 0 when x approch infinite?? so I can use L'Hopital's Rule
2. ### I Some questions on l'Hôpital rules

Hi, PF Got questions to start with: ¿some casual background about these Rules?; ¿are them two, as the textbook says?. https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule (only one statement found) Here goes the first, from "Calculus, 7th ed, R, Adams, C. Essex" THEOREM 3 The first...
3. ### Calculating Limits without L'Hopital: A Scientist's Perspective

How can I calculate preferably without L'Hopital? Thanks.
4. ### Rule de l'Hôpital - Continuous and Differentiable

Homework Statement Prove that ##h: [0,\infty) \rightarrow \mathbb R, x \mapsto \begin{cases} x^x, \ \ x>0\\ 1, \ \ x = 0\\ \end{cases} ## is continuous but not differentiable at x = 0. The Attempt at a Solution To show continuity, the limit as x approaches 0 from the right must equal to 1...
5. ### Solving Limits & L'Hopital w/ A & B Parameters

Homework Statement Determine the following limit in terms of the two real-valued parameters A and B: lim_{x \rightarrow 0} (\frac{Ae^{A/{x^2}}+Be^{B/{x^2}}}{e^{A/{x^2}}+e^{B/{x^2}}}) Homework Equations L'Hopital's rule The Attempt at a Solution I first divided by e^{A/{x^2}} in both...
6. ### Definite integral doesn't solve with L'Hopital

I want to solve: $$\int_0^\infty \frac{dx}{\left( x^2+r^2 \right)^{3/2}}=\left[ \frac{x}{r^2\sqrt{x^2+r^2}}\right]_0^\infty$$ I apply L'Hopital's to the denominator: $$\left(r^2\sqrt{x^2+r^2}\right)'=\frac{xr^2}{\sqrt{x^2+r^2}}$$ I apply again and agin L'Hopital to this but all the time almost...
7. ### Calculating Limit of sin(e^x) Using L'Hopital's Rule - Step by Step Guide

Hello (: Can someone help me calcultating this limit (as x → ∞) This is the function: If I just kept deriving sin(e^x) nothing really happened. So I tried using L'Hopital but partly, i guess: so: (1 or -1) × 1 + 0 + 0 / 0 + ∞ + ∞The answer must be 1 (according to wolfram alpha) but i...
8. ### MHB Solve Limit w/o L'Hôpital: Miras Tussupov's Q on Yaoo Answers

Here is the question: I have posted a link there to this thread so the OP can view my work.
9. ### Solve L'Hopital's Rule for y(t) at Limit of t to Infinity

So I've come across this formula that I derived. y(t) =v2t/√(v2t2+b2) I would like to solve the limit of t to infinity analytically. When I apply L'Hopital I get y = lim v2 / lim v2t/√(v2t2+b2) but as you can see I would have to apply L'Hopital rule an infinite amount of times, now I...
10. ### Limit without using de l'hopital

how can i solve lim (1+2^1/x)/(3+2^1/x) without using de l'hopital?
11. ### Is it really circular to prove the sinx/x limit with L'Hopital?

Hello, This limit \lim_{x \to 0} \frac{ \sin x}{x} is often cited as being an example where L'Hopital's rule cannot be used, since to use it you'd need to differentiate sine; but the derivative of sine, using the limit definition of a derivative, requires that you use the sinx/x limit...
12. ### Limit of Indeterminate Form (0/0) using L'Hopital Rule

Homework Statement http://puu.sh/1irk2 Homework Equations The Attempt at a Solution I am having trouble doing this question. I tried doing the L'Hopital but didn't work. When I subbed in 0 in the function, I got 0/0. first time: I got that: http://puu.sh/1iroz It still...
13. ### Proof of the L'Hôpital Rule for the Indeterminate Form

I ask for the Proof of the L'Hôpital Rule for the Indeterminate Form \frac{\infty}{\infty} utilising the Rule for the form \frac{0}{0} The Theorem: Let f,g:(a,b)\to \mathbb{R} be two differentiable functions such as that: \forall x\in(a,b)\ \ g(x)\neq 0\text{ and }g^{\prime}(x)\neq 0 and...
14. ### L'Hopital on limit of tanx(lnx) as x ->0 (from the right).

L'Hopital on limit of tanx(lnx) as x -->0 (from the right). Regarding this solution to the lim of (tanx)(lnx) as x approaches zero (from the right). I'm confused about the part I outlined in blue: What steps are going on here?
15. ### Unfamiliar algebra within L'Hopital

Homework Statement \stackrel{lim}{x\rightarrow}0^{+} (e^{2x}-1)^{x} At the step of dividing by e^x, how does that algebra work?
16. ### Indefinite forms and l'hopital

Homework Statement *indeterminate* oops the limit of x^x as x goes to zero from the right Homework Equations Going to be using L'hopital, and related algebraic manipulations to convert to indefinite form 0/0, infinity/infinity The Attempt at a Solution My understanding is that this limit...
17. ### L'Hopital Limit with rational exponent

Homework Statement I need to find the limit of this using L'Hopital Rule, if it exists. lim_{x\rightarrow0^{+}}(\frac{sinx}{x})^{1/x^{2}} Homework Equations All of our examples used lim f(x)/g(x)=f'(x)/g'(x) The Attempt at a Solution We have not dealt with any limits like this that have...
18. ### Evaluating a limit using L'Hopital

Homework Statement ForQ(x) = x^k + \sum_{n=0}^{k-1} a_n x^n Find \lim_{x→∞}\left({Q(x)^{\frac{1}{k}} - x}\right) Homework Equations L'Hopital The Attempt at a Solution Q(x) - x^k = \sum_{n=0}^{k-1} a_n x^n Q(x) - x^k = (Q(x)^{\frac{1}{k}})^k - x^k = \left(Q(x)^{\frac{1}{k}} -...
19. ### Proof l'hopital for infinity over infinity

i wonder about this proof for l'hopital for infinity over infinity: http://planetmath.org/encyclopedia/ProofOfLHopitalsRuleForInftyinftyForm.html how is this proved: http://bildr.no/view/1011658
20. ### Find lim using L'Hopital Rule

Homework Statement Find \lim_{x\to0} \frac{\arcsin(x)-x}{x^3} The Attempt at a Solution This is obviously an indeterminate form, so we apply L'hopital's rule to get \lim_{x\to0} \frac{\frac1{\sqrt{1-x^2}} - 1 }{ 3x^3} which is again an indeterminate form so we apply it again to...
21. ### A seemingly tricky limit - L'Hopital

Homework Statement \lim_{t \to \infty} \frac {1-\frac{t}{(t-1)}}{1-\sqrt{\frac{t}{(t-1)}}} Homework Equations The Attempt at a Solution I am pretty sure that everything I did here was legal, I just wanted to check. I got the right answer, so yeah, here's what I did: \lim_{t...
22. ### Mastering L'Hopital's Rule: Solving Indeterminate Forms in Calculus

Estimate the limit using L'Hopital's rule: 1. limit of x is going to infinity of xtan(1/x) Okay, I have no idea how to do #1, but i know the indetermiate form is infinity*0. 2. limit as x tends to 0+ of (lnx - ln sinx) For #2 the indeterminate form is infinity-infinity, so i took the...
23. S

### Limit of (3^(sin(t))-1)/t as t approaches 0: L'hopital's problem

Homework Statement I need to find the lim as t approaches 0 of [(3^(sin(t)))-1]/t Homework Equations The Attempt at a Solution I have no idea how to go about solving this. Whenever natural logs and "e" are involved I get confused. I know ln is involved here but I don't know how...
24. ### Find limits for x to infinity with L'Hopital

Homework Statement For the following function decide whether f(x) tends to a limit as x tends to infinity. If the limit exists find it. Homework Equations f(x)=[xsinx]/[x^2 +1] The Attempt at a Solution I thought about using L'Hopitals rule, so i got: [sinx + xcosx]/[2x]...
25. ### Can L'hopital's Rule be Applied to this Limit?

\lim_{x \rightarrow infinity}/ left(\frac{\1+Tan\frac{/pi}{2x}}{1+sin\frac{/pi}{3x}}}right)^x
26. ### L'Hospital Rule: Solving with (a) or (b)?

[SOLVED] About l'hopital rule I have two solutions for a question about limit http://tinyurl.com/2pknkb May I know is (a) correct, or (b)? What is the reason for the other to be wrong? Note: (a) just directly applies l'hopital rule to numerator (b) is to reduce the numerator using logarithm...
27. ### Understanding Arcsin and L'Hopital: Finding the Limit of (arcsin(2x))/x^3

Hey guys, have a questions about L'Hopital and arcsin. The question is to find the limit of (arcsin(2x))/x^3 as x->0. I can find the limit no problem just by applying L'Hopital, but I am having difficulty proving that it's valid to use L'Hopital with the function, because with f(x)=arcsin(2x)...
28. ### How do I solve this difficult L'Hopital problem using fractions and derivatives?

I got the following problem in my math class: http://img81.imageshack.us/img81/7508/limitdv4.jpg I know that I'm supposed to use L'Hopital's rule, but I have 2 problems. First of all, I don't know how to make that into a fraction, besides putting it all over 1 or making tan negative and...
29. ### Can we apply L'Hopital rule to 4/0 form?

1) Can we apply L'Hopital rule to the 4/0 form? eg \lim_{x\rightarrow 0} \frac{x+4}{x^2} =\lim_{x\rightarrow 0} \frac{1}{2x} =0 2) we know that 0/0 is indefinite form, but is 4/0 indefinite form?
30. ### Optimizing L'Hopital's Rule for Limits with Logarithmic Functions

I have to solve this: \lim_{\substack{s\rightarrow 0^+}} s^4 (\frac{1}{2} ln (s) - \frac{1}{8}) Here is what I did so far: \lim_{\substack{s\rightarrow 0^+}} \frac{s^4}{\frac{1}{\frac{1}{2} ln (s) - \frac{1}{8}} = = \lim_{\substack{s\rightarrow 0^+}}...
31. ### Solving Limits & Integrals with L'Hopital's Rule

Hi, I'm having trouble determining this limit: \lim_{x\rightarrow0^+}\frac{\sqrt{x}\sin\sqrt{x}}{1-e^{-x}} It's a 0/0 type of expression, so it seems like L'Hopital's rule should be applicable, but no matter how many times I differentiate I keep getting 0/0-expressions. Any hints...
32. ### Proof of L'Hospital's Rule 3: Lim f(x)/g(x) = ∞

if lim f(x)= infinity= lim g(x) x->infinity x->infinty and lim f'(x)/g'(x)=infinity x-> infinity then lim f(x)/g(x)=inifity x-> inifinity The above fact is what I am trying to prove. From my notes, i see the following: For m>0, choose k>0, such that if...
33. ### Getting derivative of the following function using L'Hopital?

Getting limit of the following function using L'Hopital? I have tried this question 5 times and each time I get a diff answer. I don't know what I am doing wrong, but here's the question. Any help? lim x-> 0 (4/x^2) - (2 / (1 - cos x)) edited - for typo in subject heading
34. ### Lim((cosx/x^2)-(sinx)/x^3)) x->0 L'hopital

Hey Everybody. I was just wondering about this query: lim((cosx/x^2)-(sinx)/x^3)) x->0 My teacher is telling me to use L'hopital, but the problem is that the first part (cosx)/(x^2) isn't a "0/0", cosx-> 1 for x->0 So what should I do, i know the right answer is -1/3, but i need...
35. ### How Does L'Hopital's Rule Apply to the Second Derivative Calculation?

Any help you can give me would be appreciated! The teacher wrote the solution to the problem on the board without giving much explanation of how he got there, and now he wants the third derivative, so I was wondering if you could help answer how he got the second. g(x) [the lim. as x goes...