Question

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n^{2}}{n^{2}+1} $$

Short answer

The limit is 1.

Step-by-step solution

Understand the Limit Expression

We need to evaluate the limit \( \lim_{n \to \infty} \frac{n^2}{n^2 + 1} \). As \( n \) approaches infinity, the expressions involving \( n^2 \) will grow large, and we need to find the behavior of this fraction.

Simplify the Fraction

We can factor \( n^2 \) from both the numerator and the denominator. This gives us \( \frac{n^2}{n^2(1 + \frac{1}{n^2})} = \frac{1}{1 + \frac{1}{n^2}} \). This simplification helps in analyzing the limit as \( n \to \infty \).

Evaluate the Limit

As \( n \rightarrow \infty \), \( \frac{1}{n^2} \) approaches zero because the denominator becomes very large. The fraction simplifies to \( \frac{1}{1 + 0} = 1 \). Thus, the limit is \( 1 \).

Key concepts

Limit Evaluation

The evaluation of limits is a fundamental concept in calculus that helps us understand how a function behaves as its variable approaches a particular point or infinity. In the exercise, we looked at the limit of a function as it approached infinity, which required us to focus on how the terms behaved when the variable grew without bounds.
Understanding the limit expression is the first step. When given a fraction like \( \lim_{n \to \infty} \frac{n^2}{n^2 + 1} \), we need to examine how both the numerator and denominator behave as \( n \) becomes very large.
In most cases, especially for rational functions, dividing every term by the highest power seen in the denominator facilitates finding the limit since it simplifies each term's behavior as the variable grows. This strategy stems from knowing that large powers of a variable will dominate smaller ones in determining a function's behavior at its extremes.

Infinite Limits

Infinite limits occur when a variable grows indefinitely, and it's essential to determine the resulting behavior of a function or expression. In the given exercise, the function \( \frac{n^2}{n^2 + 1} \) is analyzed as \( n \) approaches infinity. Here, understanding infinite limits helps us identify how the growing terms influence the entire expression.
As \( n \rightarrow \infty \), recognizing which terms become negligible versus which remain significant is crucial. For instance, concepts like \( \frac{1}{n^2} \) approaching zero are invaluable when understanding how infinite limits simplify complex expressions.
Often with infinite limits, the larger degree terms in polynomials or functions primarily dictate the outcome. Therefore, in our problem, \( n^2 \) in both the numerator and the denominator are critical. Other summands become less consequential because they don’t grow as significantly compared to the leading terms.

Limit Simplification

Simplifying limits involves reducing complex expressions into a form where evaluating the limit becomes straightforward. In the exercise, this simplification process was key.
To simplify \( \frac{n^2}{n^2 + 1} \), we factored out \( n^2 \) from the numerator and denominator, rewriting the expression as \( \frac{1}{1 + \frac{1}{n^2}} \). This form is more easily understood as \( n \) becomes very large, making the small extra term \( \frac{1}{n^2} \) negligible.
Simplifying before evaluating helps prevent errors in determining a function’s limit. It also provides a clearer picture of which terms matter most as variables approach specified inputs, especially infinities. Therefore, recognizing when and how to perform simplifications like factoring or dividing by leading coefficients is a significant skill in limit problems.