# Evaluate Limit: 7lim... (n^2 + n + 1) / (n^3 + 2n^2 +n)

• naspek
The final answer is 7.In summary, the limit of (1 + 1/n^2)(7/n+1) as n approaches infinity is 7. This is because as n gets larger without bound, the terms 1/n^2 and 7/n approach 0, leaving 1 and 7 as the remaining factors. Therefore, the limit is simply the product of 1 and 7, which is 7.
naspek

## Homework Statement

lim... (1 + 1/n^2)(7/n+1)
n->infinity

## The Attempt at a Solution

lim... (1 + 1/n^2)(7/n+1)
n->infinity

= (7/n + 1) + (7/n^3 + n^2)

bring out 7 because constant..

7lim... (1/n + 1) + (1/n^3 + n^2)
n->infinity
.
.
.
.
.
7lim... (n^2 + n + 1) / (n^3 + 2n^2 +n)
n->infinity

limit does not exist.. am i correct?

Before you try to mathematically rearrange anything, try imagining what happens to (1+(1/n^2)) as n approaches infinity. Then do the same with (7/(n+1)). What do you see?

naspek said:

## Homework Statement

lim... (1 + 1/n^2)(7/n+1)
n->infinity

## The Attempt at a Solution

lim... (1 + 1/n^2)(7/n+1)
n->infinity

= (7/n + 1) + (7/n^3 + n^2)

bring out 7 because constant..

7lim... (1/n + 1) + (1/n^3 + n^2)
n->infinity
.
.
.
.
.
7lim... (n^2 + n + 1) / (n^3 + 2n^2 +n)
n->infinity

limit does not exist.. am i correct?
No. Don't multiply the two factors. Each one has a limit that is readily obtainable. As n gets large without bound, what happens to 1/n2? What happens to 1 + 1/n2? What happens to 7/n? What happens to 7/n + 1?

The limit does exist.

## What is the purpose of evaluating this limit?

The purpose of evaluating this limit is to determine the behavior of the function as the input (n) approaches a certain value. In this case, we are interested in the behavior of the function as n approaches infinity.

## What is the general method for evaluating a limit?

The general method for evaluating a limit is to first try plugging in the value that the input is approaching. If this results in a definite value, then that is the limit. If not, then we need to use algebraic manipulation, factoring, or other techniques to simplify the expression and try again.

## How do we simplify the given expression?

To simplify the given expression, we can factor out a common term from both the numerator and denominator. In this case, we can factor out n from the numerator and n^2 from the denominator, giving us (n + 1) / (n^2 + 2n + 1). Then, we can use the fact that (a + b)^2 = a^2 + 2ab + b^2 to rewrite the denominator as (n + 1)^2. This gives us a simpler expression of (n + 1) / (n + 1)^2.

## What is the limit of this simplified expression?

The limit of this simplified expression is 1. As n approaches infinity, both the numerator and denominator approach infinity, but the denominator has a higher degree term. This means that the expression as a whole will approach 0, and the limit will be 1.

## What is the overall limit of the original expression?

The overall limit of the original expression is also 1. Since we simplified the expression to (n + 1) / (n + 1)^2, the behavior of the original expression will be the same as the simplified one. Therefore, both limits will be 1.

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