Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Rewrite the Expression

To analyze the limit \( \lim _{n \rightarrow \infty} \frac{n}{n^{2}+1} \), observe that the highest power of \( n \) in the denominator is \( n^2 \). Thus, simplify the expression by dividing both the numerator and the denominator by \( n^2 \). This gives you: \[ \frac{\frac{n}{n^2}}{\frac{n^2}{n^2} + \frac{1}{n^2}} = \frac{\frac{1}{n}}{1 + \frac{1}{n^2}}. \]

Step 2: Evaluate the Simplified Limit

Now, consider the simplified expression \( \frac{\frac{1}{n}}{1 + \frac{1}{n^2}} \) as \( n \rightarrow \infty \). As \( n \) approaches infinity, \( \frac{1}{n} \rightarrow 0 \) and \( \frac{1}{n^2} \rightarrow 0 \). Thus, the expression simplifies to \[ \frac{0}{1 + 0} = 0. \]

Key Concepts

Infinity

Infinity is a key concept in mathematics that represents a quantity larger than any attainable number. When we discuss limits, especially as a variable approaches infinity, we are interested in understanding the behavior of that variable as it grows towards an unbounded magnitude. In our example, we considered the limit as \( n \rightarrow \infty \) for a given sequence. This means we are examining what happens to the sequence's terms as \( n \) gets larger without bound. In practical terms, it's about understanding the long-term trend of the sequence. Does it approach a specific value, or does it grow indefinitely? This can help us grasp deeper insights into mathematical sequences and their properties. Infinity is not a number in the regular sense but a concept that refers to limitless potentials, either positive or negative. This can be helpful to understand phenomena in calculus, where limits often describe asymptotes and behavior approaching vast extremes.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. An example is \( \frac{n}{n^2+1} \), which we simplified in the exercise to evaluate the limit. Rational functions can have intriguing properties, especially regarding their long-term behavior. Important factors include:

• The degree of the polynomial in the numerator
• The degree of the polynomial in the denominator

These degrees determine how the function behaves as the variable approaches infinity. For instance, if the degree of the numerator is less than the denominator's degree, the rational function often approaches zero, as was the case here. Understanding rational functions is essential for solving limits, analyzing asymptotic behavior, and applying them in various mathematical and real-world contexts.

Asymptotic Behavior

The asymptotic behavior of a function describes how it acts as the input variable either becomes very large or very small. It is often used to simplify complex behaviors into understandable trends.In our exercise, examining the asymptotic behavior of \( \frac{n}{n^2+1} \) helps us determine the limit easily by identifying what happens at infinite bounds. As \( n \rightarrow \infty \), breaking down the expression revealed that the significant term in the denominator grows faster than the numerator, causing the whole fraction's value to approach zero.Asymptotic analysis lets us make insightful predictions about a function without needing exact values, merely by understanding dominant terms. This is a powerful approach in calculus for exploring limits, simplifications, and predicting system behavior in mathematical modeling.