Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{x^{2}-8 x+15}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Dominant Terms

As the variable \(x\) approaches infinity, dominant terms in the function will determine its behavior. In both the numerator and the denominator, the term \(x^2\) is dominant because it has the highest degree in\(x\). Thus, the key terms are \(x^2\) in the numerator and \(x^2\) in the denominator.

Step 2: Simplify the Expression by Factorization

To simplify, we divide every term in both the numerator and denominator by \(x^2\).\[\frac{x^2}{x^2 - 8x + 15} = \frac{x^2/x^2}{(x^2 - 8x + 15)/x^2} = \frac{1}{1 - \frac{8}{x} + \frac{15}{x^2}}\]

Step 3: Apply Limit Properties

As \(x\) approaches infinity, \(\frac{8}{x}\) approaches 0 and \(\frac{15}{x^2}\) also approaches 0 because the terms \(\frac{8}{x}\) and \(\frac{15}{x^2}\) become negligible. Thus, the expression simplifies to:\[\lim_{x \to \infty} \frac{1}{1 - \frac{8}{x} + \frac{15}{x^2}} = \frac{1}{1 - 0 + 0} = 1\]

Step 4: State the Final Limit Result

The limit of the expression as \(x\) approaches infinity is determined by the simplified expression from the previous step. Therefore, the final result is:\[\lim_{x \to \infty} \frac{x^{2}}{x^{2}-8 x+15} = 1\]

Key Concepts

Dominant Terms in Polynomials

In calculus, when dealing with limits, especially as a variable approaches infinity, dominant terms play a crucial role. A dominant term in a polynomial is the term with the highest degree. This term will have the biggest impact on the behavior of the polynomial as the variable becomes very large.

For example, in the given expression \( \frac{x^2}{x^2 - 8x + 15} \), we observe both the numerator and the denominator. The numerator has the term \( x^2 \), while the denominator has terms \( x^2, -8x, \) and \(+15\).

• The term \( x^2 \) in both the numerator and denominator is the dominant term because it is of the highest degree.

As \( x \) approaches infinity, the lower degree terms, like \(-8x\) and \(+15\), have less significance in comparison to \( x^2 \). Thus, identifying and focusing on the dominant terms helps simplify the calculation of limits.

Simplification of Rational Expressions

Simplifying rational expressions is an essential step in limit problems. By focusing on the dominant terms, we can make the expression more manageable. To do this, we divide all terms in both the numerator and denominator by the dominant term, which is \( x^2 \) in our case.

For \( \frac{x^2}{x^2 - 8x + 15} \), dividing through by \( x^2 \) gives us:

• Numerator: \( \frac{x^2}{x^2} = 1 \)
• Denominator: \( \frac{x^2}{x^2} - \frac{8x}{x^2} + \frac{15}{x^2} = 1 - \frac{8}{x} + \frac{15}{x^2} \)

This transformation simplifies the rational expression significantly.

As \( x \) approaches infinity, the terms \( \frac{8}{x} \) and \( \frac{15}{x^2} \) shrink towards zero, rendering them negligible. The expression now clearly shows the simplified form. This process highlights the importance of simplification in evaluating limits.

Limit Properties in Calculus

Limit properties are the foundational tools to solve problems where we need the behavior of a function as the variable grows extremely large or extremely small. We use these properties here to deduce the final limit.

As mentioned before, when \( x \) grows towards infinity:

• \( \frac{8}{x} \) approaches 0, because dividing a fixed number by a very large number results in a value close to zero.
• Similarly, \( \frac{15}{x^2} \) also approaches 0 even faster since the denominator, \( x^2 \), grows even larger than \( x \).

Hence, using these limit properties, the simplified expression \( \frac{1}{1 - \frac{8}{x} + \frac{15}{x^2}} \) approaches \( \frac{1}{1} = 1 \).

This demonstrates that applying the properties of limits can allow us to simplify complex expressions and arrive at the correct result with ease.