Find Limit of $\frac{n^3}{(n + 1)^2}$ as $n$ Approaches ∞

In summary, the limit of the given expression as n approaches infinity can be simplified to |n/(1+1/n)^2|, as shown by the solution. This can be derived by first rewriting the expression as |n^3/(n+1)^2| and then simplifying it further.
  • #1
tmt1
234
0
I need to find the limit of

$$\left| \frac{(n + 1)n^3}{(n + 1)^{3}} \right|$$

as $n$ approaches infinity.

I simplify this to:

$$\left| \frac{n^3}{(n + 1)^{2}} \right|$$

But the solution simplifies it to:

$$\left| \frac{n}{(1 + \frac{1}{n})^{2}} \right|$$

How do I get to this result?
 
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  • #2
tmt said:
I need to find the limit of

$$\left| \frac{(n + 1)n^3}{(n + 1)^{3}} \right|$$

as $n$ approaches infinity.

I simplify this to:

$$\left| \frac{n^3}{(n + 1)^{2}} \right|$$

But the solution simplifies it to:

$$\left| \frac{n}{(1 + \frac{1}{n})^{2}} \right|$$

How do I get to this result?
\(\displaystyle \left | \frac{n^3}{(n + 1)^{2}} \right |\)

\(\displaystyle = \left | \frac{n^2 \cdot n}{(n + 1)^{2}} \right | = \left | \frac{n}{\frac{(n + 1)^{2}}{n^2}} \right |\)

Can you finish from here?

-Dan
 

FAQ: Find Limit of $\frac{n^3}{(n + 1)^2}$ as $n$ Approaches ∞

What is the limit of the expression n3/(n + 1)2 as n approaches infinity?

The limit of the expression n3/(n + 1)2 as n approaches infinity is infinity. This means that as n becomes larger and larger, the value of the expression also becomes larger without bound.

How can I prove the limit of the expression n3/(n + 1)2 as n approaches infinity is infinity?

To prove that the limit of the expression n3/(n + 1)2 as n approaches infinity is infinity, we can use the definition of a limit. This involves showing that for any positive number M, there exists a positive number N such that whenever n is larger than N, the value of the expression is larger than M.

Is it possible for the limit of the expression n3/(n + 1)2 as n approaches infinity to be a finite number?

No, it is not possible for the limit of the expression n3/(n + 1)2 as n approaches infinity to be a finite number. As n becomes larger, the denominator (n + 1) will always be smaller than the numerator n3, resulting in a value closer to infinity.

What is the significance of finding the limit of an expression as n approaches infinity?

Finding the limit of an expression as n approaches infinity is important in many mathematical and scientific applications. It allows us to understand the long-term behavior of a function and make predictions about its values in the future. It also helps us identify any asymptotes or discontinuities in a function, which can have significant impacts on its behavior.

Can this limit be evaluated using L'Hôpital's rule?

No, L'Hôpital's rule is only applicable for limits of the form 0/0 or ∞/∞. In this case, the limit is already in its simplest form and cannot be rewritten in a way that would allow us to apply L'Hôpital's rule.

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