Taking the Limit As N -> Infinity

In summary, taking the limit as N approaches infinity involves evaluating the behavior of a function as its input values grow without bound. This concept is an important tool in calculus and is used to determine the behavior of functions at their extreme values. By observing the trend of the function as N increases, it is possible to understand the overall behavior and characteristics of the function. This limit can also be used to solve complex problems and equations in a more efficient and accurate manner.
  • #1
student45
Taking the Limit As N --> Infinity

[tex]\mathop {\lim }\limits_{n \to \infty } \frac{{n^n x}}{{(n + 1)^n }}[/tex]

Does this limit exist? Somehow it's supposed to come down to x/e
 
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  • #2
Factor out the x (the limit is with respect to n).

Consolidate the remaining factors into a term with a single exponent.

Perform a little magic inside the parentheses and look for something familiar.
 
  • #3
Got everything except the magic... the goal is to get (1+(1/n))^n, right?
 
  • #4
Well, that would be e. You want 1/e.
 
  • #5
Doh! Thanks. Got it.
 

Related to Taking the Limit As N -> Infinity

1. What does "taking the limit as N -> Infinity" mean?

When we say "taking the limit as N -> Infinity," we are referring to the mathematical process of evaluating the behavior of a function or sequence as the input or index approaches infinity. In other words, we are interested in understanding how the function or sequence behaves when the input or index becomes infinitely large.

2. How do you take the limit as N -> Infinity?

To take the limit as N -> Infinity, we use a set of rules and techniques from calculus, known as "limit laws," to evaluate the behavior of the function or sequence. These laws allow us to simplify and manipulate the expression until we reach a definitive answer or determine that the limit does not exist.

3. Why is taking the limit as N -> Infinity important?

Taking the limit as N -> Infinity is important because it allows us to analyze the long-term behavior of a function or sequence. This information is crucial in various fields of mathematics, physics, and engineering, as it helps us understand the behavior of systems that continue to change over time.

4. What are some common applications of taking the limit as N -> Infinity?

The process of taking the limit as N -> Infinity has various applications, including but not limited to analyzing the growth rate of functions, determining the convergence or divergence of infinite series, and finding the maximum or minimum values of a function. It is also essential in fields such as differential equations, calculus, and statistics.

5. Can you give an example of taking the limit as N -> Infinity?

Yes, for example, if we have the function f(x) = (3x^2 + 2x + 1) / (x^2 + 1) and we want to find the limit as x -> Infinity, we can simplify the expression by dividing each term by the highest power of x in the denominator, which is x^2. This results in f(x) = (3 + 2/x + 1/x^2) / (1 + 1/x^2). As x approaches infinity, both 2/x and 1/x^2 approach 0, and we are left with f(x) = 3/1 = 3. Therefore, the limit as x -> Infinity of f(x) is 3.

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