limit x tends to infinity ( 1+root n)/(1-rootn)
You must have meant:
lim (1 + √n)/(1-√n) as n ---->∞
divide top and bottom my √n
= lim (1/√n + 1)/(1/√n - 1) as n ---->∞
Now look at the 1/√n term
as as n ---->∞ , 1/√n -----> 0
e.g. let n = 10,000,000 , 1/√10,000,000 = .00031... getting small
So you are left with
lim (0 + 1)/(0 - 1) as n ---->∞
= -1
To find the limit of the given expression as x tends to infinity, we need to simplify the expression and observe the behavior as n (which is equivalent to x) increases.
Let's start by rationalizing the denominator:
(1 + √n)(1 - √n) = 1 - n
By canceling out the common factors (√n - √n^2), we get:
(1 + √n) / (1 - √n) = (1/n) * (1 + √n)
Now, as n approaches infinity, the term 1/n becomes infinitesimally small. Additionally, the term (1 + √n) will have a larger impact compared to 1/n. Thus, the overall expression approaches ∞.
Therefore, the limit of (1 + √n) / (1 - √n) as n tends to infinity is infinity.