https://file.aiquickdraw.com/mathbot/file/1727339975638p3c8so47.jpg

The problem is to evaluate the limit:

\[
\lim_{{n \to \infty}} \frac{(3-n)^2 + (3+n)^2}{(3-n)^2 - (3+n)^2}
\]

First, we expand the squares:

\[
(3-n)^2 = 9 - 6n + n^2
\]
\[
(3+n)^2 = 9 + 6n + n^2
\]

Substituting these expansions into the limit, we get:

\[
\lim_{{n \to \infty}} \frac{(9 - 6n + n^2) + (9 + 6n + n^2)}{(9 - 6n + n^2) - (9 + 6n + n^2)}
\]

Simplify the numerator and the denominator:

\[
\text{Numerator: } 18 + 2n^2
\]
\[
\text{Denominator: } -12n
\]

Therefore, the limit simplifies to:

\[
\lim_{{n \to \infty}} \frac{18 + 2n^2}{-12n}
\]

Since \(n^2\) grows faster than \(n\), the dominant term in the numerator is \(2n^2\) and in the denominator is \(-12n\). Hence, the limit reduces to:

\[
\lim_{{n \to \infty}} \frac{2n^2}{-12n} = \lim_{{n \to \infty}} \frac{2n}{-12} = \lim_{{n \to \infty}} -\frac{n}{6}
\]

As \(n\) approaches infinity, the limit of \(-\frac{n}{6}\) goes to \(-\infty\).

The final answer is:
\[
-\infty
\]

Do you have any questions on this, or would you like to explore further details? Here are some follow-up queries you might consider:

1. What happens if we change the limit to \(n \to -\infty\)?
2. How would the limit change if we added a constant to the numerator?
3. What is the significance of the growth rates of polynomial terms in limits?
4. Could this limit be evaluated using L'Hopital's Rule?
5. How do these types of limits apply in practical scenarios?

**Tip**: When dealing with limits involving large powers of a variable, focus primarily on the terms with the highest degree, as they tend to dominate the behavior of the function as the variable tends to infinity or negative infinity.

lim (n → ∞) [(3 - n)^2 + (3 + n)^2] / [(3 - n)^2 - (3 + n)^2]

This step-by-step solution explains how to evaluate the limit lim (n → ∞) [(3 - n)^2 + (3 + n)^2] / [(3 - n)^2 - (3 + n)^2] by simplifying polynomial expressions and analyzing their behavior at infinity. Key concepts include polynomial expansion and the impact of dominant terms as n approaches infinity. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation