This is a Calculus Question?????

Question

Show how to use analytical techniques to determine the limit: limit (x--> to infinite)= (1+1/2x)^x.

Answer 1

Is the first x in the numerator? I'll assume so since you don't have parentheses around the 2x. 1/2x is different than 1/(2x).

This limit goes to infinity because (1+x/2) is greater than one, and by it being raised to large powers makes it get exponentially larger.

If you meant the limit of (1+1/(2x))^x as x goes to infinity, then the limit is indeed 1.

Answer 2

Azanuddin at least has the right answer, but doesn't actually answer the question (how to use analytical techniques...)

Let y = (1 + 1/(2x))^x. Then ln y = x ln (1 + 1/(2x)).
Write a Taylor series for ln (1 + 1/(2x)):
ln (1 + z) = z - z^2/2 + z^3/3 - ...
So ln y = x (1/(2x) - 1/(8x^2) + O(1/x^3))
= 1/2 - 1/8x + O(1/x^2)
So as x -> ∞, ln y -> 1/2. Hence y -> e^(1/2) = √e.

Answer 3

Looks to me like the limit = 1. 1/2x goes to 0. 1 to infinity power = 1.

Answer 4

limit((1+1/2/x)^x, x = infinity)
this is the exponential functions, as x goes to large number, the power of the functions is going to extreme big so
limit((1+1/2/x)^x, x = infinity) = e^(1/2)