Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Understand the Limit Expression

The given limit expression is \( \lim_{n \to \infty} \frac{n^2}{n+1} \). We need to analyze the behavior of this expression as \( n \) approaches infinity.

Step 2: Divide by the Highest Power of n in the Denominator

The highest power of \( n \) in the expression is \( n \) from the denominator. So, we divide both the numerator and denominator of the fraction by \( n \):\[\frac{n^2}{n+1} = \frac{n^2/n}{(n+1)/n} = \frac{n}{1 + \frac{1}{n}}\]

Step 3: Apply the Limit to the Simplified Expression

Now, evaluate the limit of the expression \( \frac{n}{1 + \frac{1}{n}} \) as \( n \to \infty \):As \( n \to \infty \), the term \( \frac{1}{n} \to 0 \). Therefore, the expression simplifies to:\[\lim_{n \to \infty} \frac{n}{1 + \frac{1}{n}} = \lim_{n \to \infty} \frac{n}{1 + 0} = \lim_{n \to \infty} n\]

Step 4: Evaluate the Final Limit

The limit of \( n \) as \( n \to \infty \) is clearly infinity, as the value of \( n \) increases without bound.

Key Concepts

Infinity in Calculus

Infinity is a crucial concept in calculus, helping us understand the behavior of functions as they approach infinitely large or small values.
When we talk about limits and infinity in calculus, we are examining what happens to a function as it nears a boundary that is not finite.

• The symbol \( \infty \) stands for infinity, signifying that a number is growing beyond any finite measure.
• In mathematics, finding the limit as \( n \to \infty \) for a function often means looking at the end behavior of the function.
• In calculus, infinity is not a number, but a way to conceptualize growth beyond bounds.
• Analyzing limits at infinity helps us predict how a function behaves and whether it has an asymptotic nature.

This concept allows us to make sense of expressions like the given problem \( \lim _{n \rightarrow \infty} \frac{n^{2}}{n+1} \), where \( n \) increases without limit.

Rational Functions

Rational functions are a type of mathematical function where one polynomial is divided by another. They are of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
These functions exhibit interesting behavior as their variables approach infinity.

• The degree of the polynomial in the numerator relative to the denominator significantly impacts the behavior of the function.
• High degrees in the numerator can push the function towards infinity; high degrees in the denominator can bring it close to zero.
• Studying the limit of a rational function often involves comparing the degrees of the polynomials in the numerator and the denominator.

In our exercise, the function \( \frac{n^{2}}{n+1} \) is a rational function. By examining the degrees of the numerator and denominator, students can comprehend why the limit resolves to infinity as \( n \to \infty \).

Dividing by the Highest Power of n

To solve limits involving rational functions effectively, a key strategy is dividing by the highest power of \( n \) present in the denominator.
This technique simplifies the evaluation of the limit, especially when dealing with infinity.

• It focuses on reducing the fraction to a simpler form by considering the highest degree of \( n \) in the denominator.
• By dividing the entire expression by that highest power, balance is restored between the numerator and denominator.
• This method helps strip away lower-order terms that become insignificant as \( n \to \infty \).
• It's a strategic mathematical maneuver that allows us to glimpse the true end behavior of the function.

Applying this to our exercise, dividing each term in the fraction \( \frac{n^2}{n+1} \) by \( n \) resulted in \( \frac{n}{1 + \frac{1}{n}} \). This reduction highlights the dominant term (\( n \)) and makes the subsequent limit much easier to evaluate.