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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{\textstyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}
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{\textstyle g'(x)\neq 0}
for all x in I with x ≠ c, and
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{\textstyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
exists, then
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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.
Homework Statement
Use L'Hopital's rule to evaluate the limit:
lim (x/(x+1))^x
x>infinite
Homework Equations
The Attempt at a Solution
I put it into a logarithm first to make it the limit of xln(x/(x+1)) then I took the derivative and got [(1/x)-(1/(x+1))]/[x^-2]
but its still...
Homework Statement
(a) Explain why L'Hopital's rule does not apply to the problem
lim_{x\rightarrow0} [ (x^{2}sin(1/x)) / sinx ]
(b) Find the limit.
Homework Equations
lim _{x\rightarrow0} xsin(1/x) = 0 , by the Squeezing Theorem.
lim _{x\rightarrow0} sin (1/x) Does Not Exist...
I'm suppose to find the limit as x goes to infinity of [(9x + 1) ^ (1/2)] / [(x+1) ^ (1/2)].
L'Hopital's Rule does not work on here (it said so even in the directions) as the function keeps on cycling.
...they gave us the answer (3) but I need to find out how.
:S
I would like to know if I did these the correct way.
Homework Statement
1.lim x->0 (1/sin 2x)-1/2x. the answer is 0.
2.lim x->0 (x^-5*ln x). The answer is -infinity.
Homework Equations
1.I used L'Hopitals theorem
2.I derived them, than L'hopitals.
The Attempt at a Solution
1.I...
Im trying to solve this problem using l'hopital but amm not sure how to do it
lim
X->infinite x^3 * e^(-x^2)
soo this infinite * e^-infinite... but from there I am not sure if you can use it to solve this...
L'Hopital's Rule - I'm loosing my hair!
Ok, I have the following:
Lim x->0 sqrt(4-x^2) -2 /x
After I change the equation to remove the radicle, I get:
Lim x->0 ((4-x^2)^1/2 - 2)/x
but when I apply the rule the, I'm loosing it I thought I should get:
1/2 ((4-x^2)^-1/2 times...
Homework Statement
lim as x->0 (tan(x)-x)/(sin2x-2x)
Homework Equations
L'Hopitals rule states that if the limit reaches 0/0, you can take the derivative of the top and the bottom until you get the real limit.
The Attempt at a Solution
(sec^2(x)-1)/(2cos2x-2) still 0/0...
evaluate the limit as x=>0 of sqrt(x^2+x) - x
so i multiplied by the conjugate to get:
x^2 + x - x^2 / sqrt(x^2 +x) + x
which simplifies to:
x / sqrt (x^2 + x) + x = infinity/infinity
taking LH, you get:
1 / 1/2(x^2+x)^(-1/2) * (2x+1) + 1 = infinity/infinity again
looking...
Hey,
So I'm having a bit of difficulty with two of these L'Hospital's Rule problems... The first:
\mathop {\lim }\limits_{x \to \infty} (\sqrt{x^2 + x} - x)
So when you have an indefinite form \infty - \infty, you've got to turn it into a product indefinite form, usually something like \infty...
Hi. Use L'Hopital's Rule to evaluate the limit.
lim x-infinity of (lnx) ^(2/x)
The answer is 1.
I kept taking the derivative but it seemed like I was going around in circles. Any help would be appreciated.
Thanks
I am trying to work through the following problem:
if function is differentiable on an interval containing 0 except possibly at 0, and it is continuous at 0, and 0= f(0)= lim f ' (x) (as x approaches 0). Prove f'(0) exists and = 0.
I thought of using the definition of a limit to get to lim [...
Having trouble with a question on L'Hopital's rule. I have never come across it must have misseda lecture. From what I understand the rule approximates values at a limit. Here's what I have anyway.
I've derived a velocity gradient for a spherically symmetric, isothermal stellar wind as...