Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
What is the limit as x goes to infinity of (1+2/x)^x and how do I get it?
 2 years ago
 2 years ago
What is the limit as x goes to infinity of (1+2/x)^x and how do I get it?
 2 years ago
 2 years ago

This Question is Closed

dumbcowBest ResponseYou've already chosen the best response.1
put in indeterminate form then apply L'Hopitals rule by differentiating top and bottom rewrite it as \[\frac{(x+2)^x}{x^x}\]
 2 years ago

claudiagBest ResponseYou've already chosen the best response.0
I think I get it but, the derivation is kind of weird... or I don't really know if I'm doing it right... would it bother you too much to walk me throught it?
 2 years ago

dumbcowBest ResponseYou've already chosen the best response.1
\[y = x^{x} \rightarrow \ln y= x \ln x\] Now differentiate both sides \[\frac{1}{y}\frac{dy}{dx} = \ln x +1\] multiply by y on both sides where y = x^x \[\frac{dy}{dx} = x^{x}(\ln x+1)\]
 2 years ago

dumbcowBest ResponseYou've already chosen the best response.1
ok i realized i was wrong in my approach, those derivatives do get messy and dont help find a solution. my apologies, here is the correct solution step by step transform problem using natural log identity, e^ln(x) = x \[e ^{\lim_{x \rightarrow \infty}\ln (\frac{x+2}{x})^{x}}\] use log rules to expand and move x to denominator \[e ^{\lim_{x \rightarrow \infty}\frac{\ln (x+2) \ln x}{\frac{1}{x}}}\] here we notice if we evaluate limit we get infinf/0 which is indeterminate so apply L'hopitals rule and differentiate top and bottom \[e ^{\lim_{x \rightarrow \infty}\frac{\frac{1}{x+2}\frac{1}{x}}{\frac{1}{x ^{2}}}}\]
 2 years ago

dumbcowBest ResponseYou've already chosen the best response.1
combine fractions on top then multiply by x^2 on top and bottom \[e ^{\lim_{x \rightarrow \infty}\frac{2x}{x+2}}\] if you evaluate again you'll get infinity over infinity so differentiate again \[e ^{\lim_{x \rightarrow \infty}2}\rightarrow e ^{2}\] so the limit of the function is e^2
 2 years ago

claudiagBest ResponseYou've already chosen the best response.0
Thank's I'll try it out. I don't really understand the first step in which you transform the problem using the idientity, I don't understand how is that possible. But I do understand the resto of it so I'll find my teacher and ask him about that. Thanks a lot!!
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.