Lim (1+(a/x))^(bx)
x->infinity
I'm having a lot of trouble with this limit.
If you can use the fact that lim(1 + 1/x)^x = e, then
let u = a/x and we now have
lim(1 + 1/u)^(b*au)
= lim((1 + 1/u)^u)^ab
= e^ab
If you do a web search for "limit 1+a/x^bx" the first site shows a way to evaluate it using L'Hospital's Rule.
To evaluate the limit of the expression lim (1 + (a/x))^(bx) as x tends to infinity, we can use the concept of exponential limits.
Let's simplify the expression step by step:
1. Start by dividing both the numerator and denominator by x:
lim (1 + (a/x))^(bx) = lim ((1 + (a/x))^x)^(b)
2. Now, we'll focus on the exponent (1 + (a/x))^x as x approaches infinity. This is a well-known limit and tends to e^a:
lim ((1 + (a/x))^x)^(b) = (e^a)^(b) = e^(ab)
So, the limit of the given expression as x tends to infinity is e^(ab).
In summary, the limit of the expression lim (1 + (a/x))^(bx) as x approaches infinity is e^(ab).