Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow \infty} \frac{3 x^{3}-x^{2}}{\pi x^{3}-5 x^{2}} $$

Step-by-Step Solution

Answer

The limit is \(\frac{3}{\pi}\).

Step 1: Identify the highest power of x in the numerator and denominator

The highest power of \( x \) in the numerator \( 3x^3 - x^2 \) is \( x^3 \), and in the denominator \( \pi x^3 - 5x^2 \) is also \( x^3 \).

Step 2: Factor out the highest power of x from both the numerator and denominator

Divide each term of the numerator and denominator by \( x^3 \):\[\frac{3x^3 - x^2}{\pi x^3 - 5x^2} = \frac{x^3(3 - \frac{1}{x})}{x^3(\pi - \frac{5}{x})}\]

Step 3: Simplify the expression

Cancel out \( x^3 \) from the numerator and the denominator:\[\frac{3 - \frac{1}{x}}{\pi - \frac{5}{x}}\]

Step 4: Evaluate the limit as x approaches infinity

As \( x \to \infty \), \( \frac{1}{x} \to 0 \) and \( \frac{5}{x} \to 0 \). Therefore, the expression simplifies to:\[\frac{3}{\pi}\]

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomials. That means it has the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). Understanding rational functions involves exploring how both the numerator \( P(x) \) and the denominator \( Q(x) \) can influence the behavior of the function, particularly as \( x \) grows very large or very small.

• The degree of the numerator and denominator greatly affects the function's behavior at extreme values of \( x \).
• If the degree of the numerator equals the degree of the denominator, their leading coefficients will determine the horizontal asymptote.
• Learning how to factor and simplify such functions makes it easier to handle complex limit problems, such as the one demonstrated in this exercise.

Infinite Limits

Infinite limits occur when the function's value increases or decreases without bound as \( x \) approaches a particular value or as \( x \) approaches infinity. For rational functions, it is crucial to evaluate what happens to the function as \( x \) moves toward infinity. This often requires examining the dominant terms.

• The 'dominant term' is the term with the highest power in the polynomial, which gives insight into the function's end behavior.
• By factoring out this dominant term, you can simplify the function. After simplification, as terms with \( \frac{1}{x} \) approach zero, the limit becomes easier to evaluate.

Recognizing infinite limits can reveal the function's asymptotic behavior, which is essential for graphing rational functions and predicting their long-term trends.

Factorization in Calculus

Factorization plays a pivotal role in simplifying expressions, especially in calculus when dealing with limits and rational functions. It involves breaking down a polynomial into a product of simpler polynomials or terms. This strategy can make evaluating limits much more straightforward.

• For rational functions, factorization helps eliminate complex parts of the expressions that can obscure the behavior of the function near certain \( x \) values.
• In the exercise, by factoring out the highest power of \( x \), both the numerator and denominator were simplified, providing a clear path to calculate the limit.
• Accurate factorization allows for clean cancellation of terms across the numerator and denominator, which is crucial in converging to the correct limit.

Mastering factorization in calculus will greatly enhance your ability to solve limit problems and understand their implications.