Jbreezy said:Homework Statement
Quick question. Brain flagrance.
Homework Equations
Why limit n--> infinity (3/4)^n = 0
The Attempt at a Solution
?? Why is this. How do you know.
They are likely using a result from a previous problem. If so, the purpose of the substitution was to get the problem in that form.Jbreezy said:Got it. What about this? THanks
limit (1+k/n)^n as n->infinity
Just go here http://www.calcchat.com/book/Calculus-ETF-5e/
You have to put in chapter 9 section 1 and question 69. I just don't understand how they knew to make a substitution.
For the substitution u = k/n, if n → ∞, what does u do?Jbreezy said:Ps then after they make the sub with u they write it as u goes to 0...huh?
No, you didn't. As k/n goes to infinity, so does u, since u = k/n. The way to look at it is, as n → ∞, then the fraction k/n → 0.Jbreezy said:Stays the same? Isn't k a constant? So OH lol so you take the limit as u goes to 0 because as k/n goes to infinity u goes to 0? I say that right?
Mark44 said:Sorry, I was typing faster than I was thinking. I meant "what does u do" as n → ∞?(I edited my earlier post.)
No, you didn't. As k/n goes to infinity, so does u, since u = k/n. The way to look at it is, as n → ∞, then the fraction k/n → 0.
The limit of (3/4)^n approaching 0 as n approaches infinity is a result of exponential decay. As n increases, the value of (3/4)^n decreases significantly, approaching 0 but never actually reaching it. This trend continues infinitely, resulting in the limit of (3/4)^n approaching 0.
The value of 0 in this limit represents the asymptote, or the boundary that the function (3/4)^n approaches but never reaches. In this case, the value of 0 represents the limit of the function as n approaches infinity.
To calculate the limit of (3/4)^n as n approaches infinity, you can use the formula for exponential decay, which is lim (a^n) = 0 when a < 1. In this case, a = 3/4, so the limit of (3/4)^n as n approaches infinity is 0.
No, it is not possible for the limit of (3/4)^n to approach a value other than 0. This is because the function (3/4)^n represents exponential decay, and as n approaches infinity, the value of (3/4)^n will always decrease towards 0 but never reach any other value.
This limit can be applied to real-world scenarios such as compound interest or population growth. In these situations, the value of (3/4)^n represents the decay or decrease in the initial amount, as n (time or generations) increases. The limit of (3/4)^n approaching 0 shows that the initial amount will eventually decrease to almost nothing over time or generations.