-%2F(2n).jpeg "(2n+1) /(2n)")
The Limit of (2n+1)/(2n) as n approaches infinity
The expression (2n+1)/(2n) is a rational function that represents a sequence. As n gets larger and larger, the behavior of this sequence becomes increasingly interesting.
Simplifying the Expression
Before we delve into the limit, let's simplify the expression:
(2n+1)/(2n) = (2n/2n) + (1/2n) = 1 + (1/2n)
This simplification helps us visualize what happens as n approaches infinity.
The Limit as n approaches infinity
As n approaches infinity (n -> ∞), the term (1/2n) approaches zero. This is because the denominator grows infinitely large, making the fraction infinitely small.
Therefore, the limit of (2n+1)/(2n) as n approaches infinity is:
lim (n -> ∞) (2n+1)/(2n) = lim (n -> ∞) [1 + (1/2n)] = 1 + 0 = 1
This means that as n gets infinitely large, the value of (2n+1)/(2n) gets closer and closer to 1.
Visualizing the Limit
You can visualize this concept by plotting the graph of the function y = (2n+1)/(2n). As n increases, the graph gets closer and closer to the horizontal line y=1. This line is called the asymptote of the function.
Conclusion
The expression (2n+1)/(2n) represents a sequence that converges to 1 as n approaches infinity. This is a fundamental concept in calculus and demonstrates how limits can be used to analyze the behavior of functions as their input values become very large.