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
x
→
c
f
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x
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=
lim
x
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c
g
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=
0
or
±
∞
,
{\textstyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}
and
g
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x
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≠
0
{\textstyle g'(x)\neq 0}
for all x in I with x ≠ c, and
lim
x
→
c
f
′
(
x
)
g
′
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x
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{\textstyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
exists, then
lim
x
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c
f
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x
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g
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x
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=
lim
x
→
c
f
′
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g
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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.
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!
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}##...
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]...
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!
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...
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...
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...
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
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...
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...
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...
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:
\[\lim_{x\rightarrow 0} \frac{ln(x^{2}+e^{x})}{ln(x^{4}+e^{2x})}\]...
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...
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...
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...
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...
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...
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...
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...
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
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...
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=...
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...
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)| <...
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...
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...
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...
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...
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...
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...
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...
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...
L'Hopitals rule here makes it way more complicated. Is there a better method?
$\alpha = 2\arcsin\left(\sqrt{\frac{s}{2a}}\right)$
$\beta = 2\arcsin\left(\sqrt{\frac{s-c}{2a}}\right)$
$$
\lim_{a\to\infty}\left[a^{3/2}(\alpha - \beta -(\sin(\alpha) - \sin(\beta))\right]
$$
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
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...
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 =...
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.
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...
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?
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...
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...
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...
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...
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...
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...
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...