Question

why 2n+1/n+1 for n=infinity the answer is 2?

Answer 1

I'm assuming you are asking why,
\[\lim_{n \rightarrow \infty} \frac{2n+1}{n+1} = 2\]

Correct?

Since both the numerator and denominator go to \(\infty\) in the limit you can do L'Hopital's Rule to prove this limit. Recall that in this case this simply means taking the derivative of the numerator and denominator and the taking the limit of the result. So,

\[\lim_{n \rightarrow \infty} \frac{2n+1}{n+1}=\lim_{n \rightarrow \infty} \frac{2}{1} = 2\]

Answer 2

Another way to look at it is graphing (2x + 1)/(x + 1) and look at what happens when x is really big.

Another way to think about it is to split up the fraction.
(2n + 1)/(n+1) = (2n)/(n+1) + (1)/(n + 1).
Taking the limit of both sides as n goes to infinity,
we see that (n)/(n+1) approaches 1 as n goes to infinity,
so (2n)/(n+1) approaches 2 as n goes to infinity.
Also, 1/(n+1) approaches 0 as n goes to infinity.

So (2n)/(n+1) + (1)/(n+1) approaches 2 + 0, or 2, as n goes to infinity.