Question

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

Short answer

The limit is 1.

Step-by-step solution

Identify the Dominant Term

Observe the expression \( \frac{n}{\sqrt{n^2+1}} \). The highest power of \( n \) in both the numerator and the term inside the square root is 2. Thus, \( n^2 \) is the dominant term in the denominator.

Simplify the Expression

Rewrite the denominator to clearly identify growth rates: \( \sqrt{n^2+1} = \sqrt{n^2(1+\frac{1}{n^2})} = \sqrt{n^2}\sqrt{1+\frac{1}{n^2}} \). This simplifies to \( n \sqrt{1+\frac{1}{n^2}} \). Substitute back: \( \frac{n}{n \sqrt{1+\frac{1}{n^2}}} = \frac{1}{\sqrt{1+\frac{1}{n^2}}} \).

Find the Limit as n Approaches Infinity

Take the limit of the simplified expression: \( \lim_{n \to \infty} \frac{1}{\sqrt{1+\frac{1}{n^2}}} \). As \( n \to \infty \), the term \( \frac{1}{n^2} \to 0 \), thus the expression becomes \( \frac{1}{\sqrt{1+0}} = \frac{1}{1} = 1 \).

Key concepts

Dominant Term

In calculus, understanding the dominant term in an expression is crucial when dealing with limits as variables approach infinity. This concept applies mainly when different terms in the expression grow at different rates. For the given exercise, \(\frac{n}{\sqrt{n^2+1}}\), identifying the dominant term helps us simplify the expression effectively.

To determine the dominant term, you need to look for the term with the highest power of \(n\). Here, the dominant term in the denominator is \(n^2\). This is because \(n^2\) grows the fastest as \(n\) becomes very large. Other terms will grow comparatively slower or remain constant.

This understanding allows us to simplify the expression by focusing on the dominant term, helping us evaluate the behavior of the function as \(n\) approaches infinity.

Infinity

In the context of limits, infinity represents an unbounded behavior of a function or sequence. As \(n\) approaches infinity, we want to understand what the expression does when \(n\) gets very large. The infinite limit concept helps in analyzing such behavior accurately.

Consider the simplified expression \(\frac{1}{\sqrt{1+\frac{1}{n^2}}}\). As \(n\) grows larger and larger, the term \(\frac{1}{n^2}\) becomes exceedingly small, approaching zero. This simplifies the expression inside the square root to \(\sqrt{1+0}\), which is just 1. Therefore, the limit of the entire expression as \(n\) approaches infinity is \(\frac{1}{1} = 1\).

This approach helps illustrate how reflecting on the behavior at infinity allows us to determine the limit of a complex expression.

Simplifying Expressions

Simplifying mathematical expressions is an essential skill, especially when evaluating limits. In this exercise, simplifying \(\frac{n}{\sqrt{n^2+1}}\) involves rewriting and rearranging terms to make the calculation more intuitive.

The process begins by rewriting the square root in the denominator: \(\sqrt{n^2+1}\). This is expressed as \(\sqrt{n^2(1+\frac{1}{n^2})}\), which further simplifies to \(n\sqrt{1+\frac{1}{n^2}}\). This simplification unveils the growth rates of each component in the expression.

Finally, substituting this back gives \(\frac{n}{n\sqrt{1+\frac{1}{n^2}}}\), which simplifies to \(\frac{1}{\sqrt{1+\frac{1}{n^2}}}\). This reduction eliminates the variable \(n\) in the numerator, emphasizing how significantly simplification can aid in determining the limit efficiently.