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: Analyze the Expression

The expression is \( \lim _{n \rightarrow \infty} \frac{n}{n^{2}+1} \). To find the limit, we need to analyze how both the numerator and the denominator behave as \( n \) approaches infinity.

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

The denominator \( n^2 + 1 \) has the highest power of \( n \) as \( n^2 \). We will divide both the numerator and the denominator by \( n^2 \). This gives us \( \frac{\frac{n}{n^2}}{\frac{n^2 + 1}{n^2}} = \frac{\frac{1}{n}}{1 + \frac{1}{n^2}} \).

Step 3: Apply the Limit

Now, apply the limit \( n \rightarrow \infty \) to the expression \( \frac{\frac{1}{n}}{1 + \frac{1}{n^2}} \). As \( n \rightarrow \infty \), \( \frac{1}{n} \rightarrow 0 \) and \( \frac{1}{n^2} \rightarrow 0 \). Thus, the expression becomes \( \frac{0}{1 + 0} = 0 \).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to find limits that resulted in indeterminate forms. It can be particularly useful when dealing with ratios of functions where both the numerator and the denominator approach zero (0/0) or infinity (∞/∞) as the variable approaches a certain value. L'Hôpital's Rule states:

• If \( \lim_{x \to c} \frac {f(x)}{g(x)}\) yields an indeterminate form, and provided that the derivatives \( f'(x) \) and \( g'(x) \) exist and \( g'(x) eq 0 \), then
• \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), assuming the limit on the right-hand side exists or is infinite.

To apply L'Hôpital's Rule, you should identify if the situation is giving an indeterminate form such as 0/0 or ∞/∞, then check whether you can differentiate the numerator and denominator. Remember, L'Hôpital’s Rule can be applied repeatedly if the resulting limit again gives an indeterminate form.

Indeterminate Forms

Indeterminate forms arise in calculus as a situation when a direct application of limits does not clearly resolve to a specific number, typically due to the form of the expression. Some common indeterminate forms include:

• 0/0
• ∞/∞
• 0 × ∞
• ∞ - ∞
• 1^∞
• 0⁰
• ∞⁰

These forms signal that further algebraic manipulation or calculus methods, such as L'Hôpital's Rule, are necessary to determine the exact behavior of the limit. Recognizing an indeterminate form is crucial for correctly addressing limits and often involves transforming the expression or using derivatives for simplification.

Infinity in Calculus

In calculus, infinity (∞) is a concept describing quantities larger than any finite value. It often occurs when discussing limits as a variable grows without bound. There are a few important behaviors to note:

• As \( n \to \infty \), \( \frac{1}{n} \to 0 \) because the fraction becomes smaller.
• Polynomials and other functions can tend towards positive or negative infinity as their degree dictates their limiting behaviors.
• Infinity is not a number but a concept; it enables the discussion of growth beyond fixed bounds.

While working with infinity in calculus, it is important to understand how terms behave and dominate each other, for instance, higher-order terms (like \( n^2 \)) will dominate lower-order terms (like \( n \)) as the variable tends towards infinity. This helps in simplifying and evaluating limits appropriately.

Rational Functions in Calculus

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. The behavior of such functions as \( x \) approaches infinity is often evaluated using limits. To analyze these functions:

• Determine the degrees of the polynomials in the numerator and the denominator. The degree is the highest power of \( x \) in the polynomial.
• If the degree of \( P(x) \) is less than \( Q(x) \), the limit is 0.
• If the degrees are equal, the limit is the ratio of the leading coefficients.
• If the degree of \( P(x) \) is greater than \( Q(x) \), the limit is ±∞, depending on the leading terms and their signs.

Evaluating limits involving rational functions is greatly aided by these rules. For following the limit process, sometimes dividing each term by the highest power of \( x \) present in the denominator can simplify the expression.