Limit question....?

Question

Let f be a continuous function. Find

lim x-->infinity f((1 - 1/x)^x)

Answer 1

Since f is continuous, then lim x->infinity f((1 - 1/x)^x) = f(lim x->infinity ((1 - 1/x)^x). Let's find the inside limit.

lim x->infinity ((1 - 1/x)^x) = lim x->infinity (e^(x ln (1 - 1/x))) = e^(lim x->infinity (x ln (1 - 1/x))). Let's find the insidemost limit.

lim x->infinity (x ln (1 - 1/x)) = lim x->infinity (ln (1 - 1/x))/(1/x). Now use L'Hospital's Rule.

lim x->infinity ((1/(1 - 1/x)) * 1/x^2)/(-1/x^2) = lim x->infinity (-1/(1 - 1/x)) = -1/(1 - 0) = -1.

We now know the limit of x ln (1 - 1/x), so we can find the limit of f((1 - 1/x)^x):

lim x->infinity f((1 - 1/x)^x) = lim x->infinity f(e^(x ln (1 - 1/x))) = f(e^(-1)) = f(1/e).

Answer 2

I think the answer should be 1.
If you plug in infinity for x, the 1/infinity will become infinitely small, and the answer you would receive from (1- 1/infinity) will get closer and closer to 1 as x approaches infinity. So you basically end up with 1^infinity which will be 1.

Answer 3

lim x-->infinity f((1 - 1/x)^x) = f(lim x-->infinity ((1 - 1/x)^x)) because f is continuous
and lim x-->infinity ((1 - 1/x)^x)= e
therefore
lim x-->infinity f((1 - 1/x)^x) = f(e) .

Answer 4

lim x-->infinity f((1 - 1/x)^x) = f(1/e)