L'hopital's rule Definition and 115 Threads

  1. M

    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. Mohmmad Maaitah

    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. murshid_islam

    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. MrNotknowinganything

    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. murshid_islam

    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. K

    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. C

    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. akaliuseheal

    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. H

    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. EEristavi

    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. B

    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. J

    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. A

    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. T

    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. T

    I How Does L'Hopital's Rule Intuitively Explain Limits?

    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. Y

    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. R

    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...
  18. S

    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. iwantcalculus

    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. rayne1

    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. Math Amateur

    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. M

    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. Maged Saeed

    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. S

    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. A

    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. A

    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. Greg Bernhardt

    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. S

    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. P

    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. F

    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. M

    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. J

    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. Q

    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. Mandelbroth

    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. F

    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. S

    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. D

    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. Petrus

    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. MarkFL

    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. D

    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. C

    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. U

    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. C

    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. N

    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

    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. P

    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. R

    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. C

    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. S

    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. M

    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...
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