Question

Limit X tends to infinity

(X+C/X-C)^x=4

Then find C

Answer 1

First of all, we need to admit that c is finite, else it makes no sense.
Then perform these steps:

Lim (x→∞) [(x-c+2c)/(x-c)]^x
= Lim (x→∞) [1+2c/(x-c)]^x

Now let: 2c/(x-c) = 1/t ==> x = c(2t+1), with t still approaching infinity. The limit becomes:

Lim (t→∞) [1+1/t]^c(2t+1) = Lim (t→∞) [(1+1/t)*(1+1/t)^2t]^c

We know that (for c finite)

Lim (t→∞) (1+1/t)^c = 1 and
Lim (t→∞) [(1+1/t)^2t]^c = Lim (t→∞) [(1+1/t)^t]^2c = e^(2c)

Therefore we need to have:

e^(2c) = 4
2c = ln4
c = ln4/2