Question

find the lim as x approaches infinity of ((1) + (1/x))^kx where k is a real number.

Hint: lim as x approaches infinity of ((1) + (1/x))^x = e

I am leaning towards e^k but that's just a guess. How should I go about starting this problem?

Answer 1

\[\left(1+\frac{1}{x}\right)^{kx}=\left[\left(1+\frac{1}{x}\right)^{x}\right]^k\]

Answer 2

so that equals e^k?

Answer 3

\[\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{kx}=\lim_{x\to\infty}\left[\left(1+\frac{1}{x}\right)^{x}\right]^k=\left[\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}\right]^k=e^k\]

Answer 4

thank you so much Zarkon... would we apply the same use of exponents for something like this: lim from x to infinity of (1 + k/x)^xkprime where k is a real number?

Answer 5

\[\lim_{x\to\infty}\left(1+\frac{k}{x}\right)^x\]

let \(u=\frac{x}{k}\) then \(\frac{1}{u}=\frac{k}{x}\) and \(ku=x\)

Assume k>0

\[\lim_{x\to\infty}\left(1+\frac{k}{x}\right)^x=\lim_{u\to\infty}\left(1+\frac{1}{u}\right)^{ku}=e^{k}\]


similarly for k<0


if k=0 then we get \(1=e^{0}\)