How do you find the limit of (n^4+6n^2)/(n^4-2n+3) as n approaches infinity?

Jul 23, 2016

lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = 1

Explanation:

Since this is also known as an infinite limit, and both the numerator and denominator are polynomials, we can divide each term by the highest power, namely by ${n}^{4}$.

Therefore, we have

lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = lim_(n->∞) (cancel((n^4/n^4))+(6n^2)/(n^4))/(cancel((n^4)/(n^4))-(2n)/(n^4)+(3)/(n^4))

= lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4))

Also, since we know that lim_(n->∞) 1/(n^m) = 0 iff m>0, we then have

lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4)) = 1/(1) = 1

We can also graph our original function to see if this limit makes sense.

graph{(x^4+6x^2)/(x^4-2x+3) [-10, 10, -5, 5]}

In this case, we can see that as n->∞, our function tends to $1$.
The text passage contains a mathematical question about finding the limit of a function as n approaches infinity, and it provides a detailed answer to this question, including the final result and an explanation of the steps taken to reach the solution.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How do you find the limit of (n^4+6n^2)/(n^4-2n+3) as n approaches infinity?

Jul 23, 2016

lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = 1

Explanation:

Since this is also known as an infinite limit, and both the numerator and denominator are polynomials, we can divide each term by the highest power, namely by ${n}^{4}$.

Therefore, we have

lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = lim_(n->∞) (cancel((n^4/n^4))+(6n^2)/(n^4))/(cancel((n^4)/(n^4))-(2n)/(n^4)+(3)/(n^4))

= lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4))

Also, since we know that lim_(n->∞) 1/(n^m) = 0 iff m>0, we then have

lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4)) = 1/(1) = 1

We can also graph our original function to see if this limit makes sense.

graph{(x^4+6x^2)/(x^4-2x+3) [-10, 10, -5, 5]}

In this case, we can see that as n->∞, our function tends to $1$.
user
The text passage contains a mathematical question about finding the limit of a function as n approaches infinity, and it provides a detailed answer to this question, including the final result and an explanation of the steps taken to reach the solution.

TYPE: SINGLE_QUESTION_WITH_ANSWER