You have x = 1/u,Painguy said: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.
lim
x→0+ (1 + x)^(1/x)
x=1/u
since x-> 0 that means 1/u ->inf
lim
x→0+ (1 + 1/u)^(1/(1/u))
= lim
1/u→∞ (1 + 1/u)^(u) = e
I feel like my last 2 steps are wrong, but I am sure my answer is right.
SammyS said:You have x = 1/u,
so if x → 0+, then so does 1/u → 0+.
What that implies is that u → +∞ .
L'Hopital's Rule is a mathematical tool used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the quotient of two functions is indeterminate, then the limit of the quotient of their derivatives will be the same. This rule can be applied to the limit definition for e, which uses the ratio of (e^x - 1)/x as x approaches 0.
L'Hopital's Rule should only be used when the limit definition for e is indeterminate, meaning that plugging in the value for x results in an undefined expression.
The steps for using L'Hopital's Rule to evaluate the limit definition for e are as follows: 1) Determine if the limit is indeterminate by plugging in the value for x. 2) If it is indeterminate, take the derivative of both the numerator and denominator. 3) Evaluate the limit using the new quotient. 4) Repeat this process until the limit is no longer indeterminate.
No, L'Hopital's Rule can only be used when x approaches 0 in the limit definition for e.
Yes, there are some limitations to using L'Hopital's Rule. It should only be used when the limit is indeterminate, and it may not always give the correct answer if used incorrectly. Additionally, it is not applicable to all types of limits, such as limits involving exponentials or logarithms.