# What is L'hopital's rule: Definition and 115 Discussions

In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital],
English: , loh-pee-TAHL) provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if

lim

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=

lim

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0

or

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{\textstyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}
and

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0

{\textstyle g'(x)\neq 0}
for all x in I with x ≠ c, and

lim

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c

f

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{\textstyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
exists, then

lim

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g
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=

lim

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{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.

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1. ### L’Hôpital’s Rule for indeterminate powers

For this, Does someone please know why we are allowed to swap the limit as x approaches zero from the right of y with that of In y? Thank you for any help!
2. ### 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
3. ### I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##

I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check? Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
4. ### Derivatives & L'Hôpital's Rule Explained

Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything
5. ### Verify a limit using L'Hopital's Rule

I have to prove that \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]= a (in order to show that the indeterminate form of the type 0^0 can be any positive real number). This is what I did: Let y = \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]...
6. ### Understanding L'Hôpital's rule

My attempt: ##\frac{f'(a)}{g'(a)} ## = ##\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\cdot\frac{h}{g(a+h)-g(a)}## = ##\lim_{h\to 0}\frac{f(a+h)-f(a)}{g(a+h)-g(a)}## I don't think I am doing this right. I don't even understand how I am supposed to use the boundary rules. I really appreciate some help!
7. ### Can L'Hopital's Rule be used to solve (0/0)^infinity limits?

It is of the form (0/0)^infinity. I know how to solve 1^infinity form
8. ### L'Hopital's Rule: Solving Homework Statement

Homework Statement Can I use L'Hopital's rule here. What I get as a solution is -30/-27 while in the notebook, without using the L'Hopital's rule the answer is -(2/27) The attempt at a solution The derivatives i get are: x/(x2+5)½ (3x2+2x)/3(x3+x2+15)⅓ 2x-5 ½ and ⅓ are there because it's...
9. ### Find limit x to infinity from f(x) contains squareroot of x

Homework Statement ##\lim x \to \infty \frac{\sqrt{x+1} - \sqrt{x}}{\sqrt{3x + 5} - \sqrt{3x + 1}}## Homework EquationsThe Attempt at a Solution ##\lim x \to \infty {\sqrt{x+1} - \sqrt{x}} * \lim x \to \infty \frac{1}{\sqrt{3x + 5} - \sqrt{3x + 1}}## ##\lim x \to \infty \frac{(x+1) -...
10. ### Calculation of limit. L'Hopital's rule

Problem: Evaluate lim(x->0) x cotx My attempt: lim(x->0) x cotx = lim(x->0) x cosx / sinx = lim(x->0) cosx * lim(x->0) x / sinx = 1 * lim(x->0) x / sinx = lim(x->0) x / sinx P.S. I know I must/can use L'Hopital's rule to evaluate indeterminate limits, but no matter how many times I derive...
11. ### Need help finding the limit of a function

Homework Statement Calculate limit as x approaches infinity of (e^x - x^3) Homework Equations ln e^x = x e^(ln x) = x The Attempt at a Solution I tried substituting x = ln e^x and got (e^x - (ln e^x)^3). I'm pretty much lost and this is my only attempt so far. I'm thinking that this is an...
12. ### L'Hopital's Rule: Homework Statement

Homework Statement Just quickly, can you apply l'hopital's rule when the limit is evaluated as undefined/undefined as in the following limit: Homework EquationsThe Attempt at a Solution
13. ### Arctan limit (without L'Hopital's Rule)

Homework Statement Limx-->positive infty arctan(1+x)/(1-x) Homework EquationsThe Attempt at a Solution I just need to know if my answer is right. Knowing that when the leading coefficients of the x when its the same, then the answer is just the ratio. So it would be -1. Then in my calculator...
14. ### MHB Why can't I apply L'Hopital's rule to all indeterminate forms?

I have a certain set of problems (i.e. https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html), where many problems are in an indeterminate form ($\frac{0}{0}$) but if we apply L'Hopital's rule it yields an incorrect answer. Instead, I have to simplify the...
15. ### I Why does l'hopital's rule work?

What is the intuition behind it? when i watch videos of people using l'hopital's rule. i can only deduce that they're only taking derivatives over and over again until a number comes out and that becomes the limit. how can a tangent slope be a value for a limit? Please give me an intuitive...
16. ### MHB Calculating Limits without Cheating or L'Hopital's Rule

Hello all, I am trying to calculate the following limits, without cheating and using a calculator (by setting a very close value of the required value of x). And no l'hopital's rule either if possible :-) The limits are: $\lim_{x\rightarrow 0} \frac{ln(x^{2}+e^{x})}{ln(x^{4}+e^{2x})}$...
17. ### MHB Solving Limits: l'Hopital's Rule & Degree Rule

Hi, I'm having some trouble with finding the limit for this question: I can use the l'hopital's rule which I tried.. I tried pi, 2pi, 0, inf, none seem to work so if I could have some help that would be appreciated! limx→0 \frac{cos5x-cos6x}{x^2} Would the degree rule apply here? It wouldn't...
18. ### How do I prove that both are equivalent limits

Homework Statement If k is a positive integer, then show that ##\lim_{x\to\infty} (1+\frac{k}{x})^x = \lim_{x\to 0} (1+kx)^\frac{1}{x}## Homework Equations L'Hopitals rule, Taylor's expansion The Attempt at a Solution How should I begin? Should I prove that both has the same limit, or is...
19. ### Solving a limit by l'hopital's rule

So, according to answer sheet, the answer is 1... The question is : limit as x approaches infinity of : squareroot( x^2 + x ) - squareroot( x^2 - x) I tried to put it in a limit calculator, but the steps shown are very complex and don't even involve l'hopital's rule... I think the solution...
20. ### MHB Improper integral and L'Hôpital's rule

integral from 2 to infinity dx/(x^2+2x-3) I got this as the result: lim x to infinity (1/4)(ln|x-1|-ln|x+1|+ln|5|) Then I got (1/4)(infinity - infinity + ln|5|) so do I need to use l'hopital's rule for ln|x-1|-ln|x+1| or would the final answer be ln|5|/4? If not, I am unsure of how to...
21. ### MHB L'Hopital's Rule _ Statement of Theorem (Houshang H. Sohrab)

I am reading Houshang H. Sohrab's book: Basic Real Analysis (Second Edition). I need help with an aspect of Sohrab's statement of Theorem 6.5.1 (L'Hopital's Rule) on pages 262-263. Sohrab's statement of Theorem 6.5.1 reads as follows: https://www.physicsforums.com/attachments/3936 At the...
22. ### Quick questions on L'Hopital's rule

Homework Statement $$\lim_{x\rightarrow -2^-}\frac{x}{x^2+x-2}$$ 2. The attempt at a solution Clearly, when graphing the above equation, the limit does not exist (or approaches positive infinity). However, when applying L'Hopital's Rule, we have $$\frac{1}{2x+1}$$ and then we can go ahead and...
23. ### Using L'Hopital's Rule to Evaluate Limits

Homework Statement How can I fount the following limit using L'H'opetal rule? $$\lim_{x\rightarrow+\infty}(ln(2x)-ln(x-4))$$ Homework EquationsThe Attempt at a Solution I tried to use the low $$\lim_{x\rightarrow a} f(x)=\lim_{x\rightarrow a}e^{ln(f(x))}=e^l$$ But it seems to be useless Many...
24. ### MHB Series Expansion & l'Hopital's rule

I'm preparing for my exam and have stumbled across this question. I understand how to execute the l'hopital's rule part of this but I just can't get there. I have no idea how to approach this in order to get a suitable series expansion to the 4th degree of x. Thank you
25. ### MHB Differentiation With L'Hopital's Rule

Hey guys, Need some more help again. I'll keep it brief. This thread is only for question 2ab. Please ignore question 1: For 2a, I simply employed L'Hopital's Rule since 0/0 is indeterminate form. Thus, my final answer came out to be: ln6-ln3. As for 2b, I computed an indeterminate form...
26. ### MHB Finding Limits Using Limit Laws (Not L'Hopital's Rule)

Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For a, I used the subtraction limit lawto get lim g(x) and lim f(x) and subtracted the answers accordingly. Then I substituted h=...
27. ### How Does L'Hôpital's Rule Solve Indeterminate Forms in Calculus?

Definition/Summary L'Hôpital's (or l'Hospital's) rule is a method for finding the limit of a function with an indeterminate form. Equations If the expression \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} has the form 0/0 or \infty / \infty, then l'Hôpital's rule states...
28. ### What is the role of the chosen value t in the proof of L'Hôpital's Rule?

I am somewhat confused by the proof of L'Hôpital's Rule in Pugh's "Real Mathematical Analysis." (See Attachments, Theorem 6). I follow every bit of the proof save one choice and its implication. That is, why choose t based on ##\displaystyle |f(t)| + |g(t)| <...
29. ### L'hopital's rule, indeterminate forms

Homework Statement lim_{x -> \infty} \left( \frac{x}{x+1} \right) ^ {x} The Attempt at a Solution So I did e^whole statement with ln(x/(x+1))*x, after that I multiplied that expression by 1/x/1/x, then I go ln(x/(x+1)/1/x, I tried taking derivative of top and bottom but it doesn't help with...
30. ### L'Hôpital's rule and weird expression

Homework Statement Find, using l'Hôpital's rule: lim x-> ∞ ((tan(ax)-atanx)/(sin(ax)-asinx)) Where a is a non constant greater than ± 1. Homework Equations -The Attempt at a Solution I can't work out anything from this point. I don't know how to change or factorize the expression (is there a...
31. ### Finding limits without use of l'Hôpital's rule.

Homework Statement Hi. I have a problem finding the limit of two different problems - without use of l'Hôpital's rule. I only know how to do this with use of the l'Hôpital's rule, therefore I'm seeking help to solve this problem.Homework Equations The problems are: Determine the limits...
32. ### Can I apply L'Hopital's rule to this integral expression?

Hi, Suppose I have \lim_{r\to 0} \left\{\int_0^{\pi} \frac{f(r,t)}{r^2}dt - \int_0^{\pi} \frac{g(r,t)}{r} dt\right\} and both integrals tend to infinity. So I combine them: \lim_{r\to 0} \int_0^{\pi} \frac{f(r,t)-r g(r,t)}{r^2} dt now at this point, the numerator in the...
33. ### Limit - Squeeze Theorem or L'Hopital's Rule?

Homework Statement Homework Equations Squeeze theorem: set up inequalities putting the function of interest between two integers. L'Hopital's rule: when plugging in the number into the limit results in a specified indeterminate form such as 0/0 or infinity/infinity then take the...
34. ### Can l'Hopital's Rule Be Generalized for Differentiable Maps Between Manifolds?

I had a wild thought. Out of curiosity, is anyone aware of a kind of generalization for l'Hopital's Rule from analysis for differentiable maps between differentiable manifolds? I'm having trouble formulating if I could do it or not, because (as far as I know), if I have ##f,g:M\to N##, with...
35. ### L'Hopital's rule for increasing functions

L'Hopital's rule greatly simplifies the evaluation of limits of indeterminate forms, especially those with polynomial terms. This is because every time you take the derivative of a polynomial, the exponent decreases by 1, until it becomes a constant function, at which point the limit can be...
36. ### L'Hopital's Rule: Evaluating Limits

Homework Statement Use l'Hopital's rule to evaluate the following limit: lim x→0 e^(-1/x) / x for x> 0. Homework Equations differentiate the top and bottom until a limit can be found. Possibly rewrite as a product. The Attempt at a Solution I was under the impression that...
37. ### MHB Is there a better way than L'Hôpital's rule?

L'Hopitals rule here makes it way more complicated. Is there a better method? $\alpha = 2\arcsin\left(\sqrt{\frac{s}{2a}}\right)$ $\beta = 2\arcsin\left(\sqrt{\frac{s-c}{2a}}\right)$ $$\lim_{a\to\infty}\left[a^{3/2}(\alpha - \beta -(\sin(\alpha) - \sin(\beta))\right]$$
38. ### MHB Solving $\frac{\sin(x-1)}{x^2+x-2}$ Without L'Hopital's Rule

Hello, I would like to solve this without lhopitals rule aswel( i succed get the answer 1/3 with lhopitals rule but do not go well without) $$\lim_{x \to 1}\frac{\sin(x-1)}{x^2+x-2}$$ Any tips i would like to have
39. ### MHB Michael's question at Yahoo Answers involving L'Hôpital's rule

Here is the original question: How to Find Lim x -> 0 of Tan(x)^x with the L'Hopital Rule? - Yahoo! Answers I have posted a link to this topic so the OP can find my response. We are given a limit to evaluate, so let's assume it exists, and write: $\displaystyle \lim_{x\to0}\tan^x(x)=L$ Take...
40. ### Indeterminate Forms and L'Hopital's Rule

Homework Statement lim ln(x-1)/(x2-x-4) x->2 Homework Equations The Attempt at a Solution Well, I thought that every time I had answers as 0/0, 2/0 or 0/2, for instance, they would constitute as indeterminate forms. I have the answer sheet for this problem. It says "answer: 0/-2 =...
41. ### Without using L'Hopital's rule, how can I calaculate this limit?

Without using L'Hopital's rule how can I calculate the limit of this function: (xn-an)/(x-a) when x→a I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.
42. ### Finding the limit without L'Hôpital's rule

Homework Statement Required to prove that \displaystyle\lim_{n\rightarrow \infty} ((1 - \frac{1}{n^2})^{n}) = 1 Homework Equations \displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n}) is bounded above by e. I'm not sure if this is relevant, but it was the first part of...
43. ### Limit without using L'Hopital's rule.

Homework Statement \lim_{x\rightarrow 1} {\frac{\sqrt[3]{x}-1}{\sqrt{x}-1}} I can do this very easily using L'Hopital's rule but in the textbook I'm going through it is a problem given before L'Hopital's rule is taught. Is there a way of doing this without using L'Hopita'ls rule?
44. ### Find limit using L'Hopital's rule, ln, and e: how did they do these steps?

Homework Statement This is from an online answer, and I don't understand the steps that it took. How did they go from the first red box to the second red box? Homework Equations L'Hopital's rule Laws of exponents The Attempt at a Solution I am really really confused. It...
45. ### A question about the L'Hopital's rule

This is how L'Hopital's Rule is defined in our textbook: "Let s signify a, a^+, a^-, \infty or - \infty and suppose f and g are differentiable functions for which hte following limit exists: lim_{x \rightarrow s} \frac{f'(x)}{g'(x)} = L ..................(1) If lim_{x...
46. ### Use L'Hopital's Rule to relate to limit definition for e

Homework Statement It can be shown that lim n→∞(1 + 1/n)^n = e. Use this limit to evaluate the limit below. lim x→0+ (1 + x)^(1/x) Homework Equations The Attempt at a Solution So i guess what i need to do is try to get that limit in the form of the limit definition for e...
47. ### Indeterminate Forms and l'Hopital's Rule

Homework Statement Lim as x→∞ of ((2x+1)/(2x-1))^(sqrtx)Homework Equations The Attempt at a Solution When I initially plugged in ∞ for my x, I get (∞/∞)^∞, correct? If so, should I just let y=((2x+1)/(2x-1))^(sqrtx) and take the limit of both sides using ln? That's what I attempted to do...
48. ### Limit at infinity; l'hopital's rule not working as expected

Homework Statement Lim(t->(inf)) 1/2((t^2)+1) + (ln|(t^2)+1|)/2 - 1/2 Homework Equations N/A (unless L'Hopital's rule can be counted as an equation for this section) The Attempt at a Solution Background: The problem started with: inf ∫(x^3)/((x^2)+1)^2 dx 0 Using partial fraction...
49. ### Question About Finding Limits with L'Hopital's Rule

Homework Statement A question we had for homework was: Which of the following is equivalent to limh->0 [arcsin((3(x+h))/4) – arcsin (3x/4)]/h ? Homework Equations There were multiple answer choices, but the correct answer is 3/(√16-9x^2).The Attempt at a Solution We've already walked through...
50. ### Limit involving L'Hopital's Rule

Homework Statement \displaystyle\lim_{x\rightarrow 0^+} x^{1/x} The Attempt at a Solution y = \displaystyle\lim_{x\rightarrow 0^+} x^{1/x} lny = \displaystyle\lim_{x\rightarrow 0^+} \frac {lnx}{x} This gives an indeterminate form and it's a quotient, so I can apply L'Hopital's Rule...