find the lim as x approaches infinity of ((1) + (1/x))^kx

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gadav478

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Hi all-

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

He gave us a hint but I don't know how to go about this problem.

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?
 
gadav478 said:
Hi all-

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

He gave us a hint but I don't know how to go about this problem.

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?
Click to expand...
You shouldn't have to "guess"!

A^(kx) = (A^x)^k by properties of exponents. Then

((1) + (1/x))^kx = [((1) + (1/x))^x]^k

. . .
 
DrPhil said:
You shouldn't have to "guess"!

A^(kx) = (A^x)^k by properties of exponents. Then

((1) + (1/x))^kx = [((1) + (1/x))^x]^k

. . .
Click to expand...


So 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?

What I have is (e^k)^kprime...
 
gadav478 said:
So 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?

What I have is (e^k)^kprime...
Click to expand...
That doesn't feel right to me. Try substitution
......y = x/k
and then look at (1 + (1/y))^y
 
I have no idea what your "prime" is supposed to indicate. Using the substitition Dr Phil suggested, you have the limit (1+ 1/y)^y. He probably thought you would recognize that limit.
 
gadav478 said:
I'm confused. With your sub, does that mean that k/x = xkprime?
Click to expand...
(1 + k/x)^(x k')
Click to expand...
If you substitute y = x/k,
then x = k y,......k/x = 1/y,
and x k' = y (k k').

Also the \(\displaystyle \displaystyle \lim_{x \to 0}\) becomes \(\displaystyle \displaystyle \lim_{y \to 0}\)
 
gadav478 said:
Hi. I am not a bot. Is it against the rules to post a question on a different site? Just wondering.
Click to expand...

There is no rule against it. The information is supplied for the benefit of the volunteer tutors - so that they do not waste time trying answer questions that has been answered at a different site.
 
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