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

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


limit does not exist.. am i correct?

 

[mg0stisha]

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?

 

[Mark44]

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.