Problem 11 - Thomas Calculus

Question

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow \infty} \frac{5 x^{3}-2 x}{7 x^{3}+3} $$

Step-by-Step Solution

Answer

The limit is \(\frac{5}{7}\).

Step 1: Verify Indeterminate Form

First, check if the limit as \(x\) approaches infinity is in the indeterminate form \(\frac{\infty}{\infty}\). Both the numerator \(5x^3 - 2x\) and the denominator \(7x^3 + 3\) tend towards infinity as \(x\) becomes very large. Thus, this is an indeterminate form suitable for l'Hopital's Rule.

Step 2: Apply l'Hopital's Rule

According to l'Hopital's Rule, if the original limit \(\frac{f(x)}{g(x)}\) gives \(\frac{\infty}{\infty}\), the limit can be found by evaluating \(\lim_{x \to \infty} \frac{f'(x)}{g'(x)}\). Differentiate the numerator: \(f'(x) = 15x^2 - 2\) and the denominator: \(g'(x) = 21x^2\).

Step 3: Find the New Limit

Calculate the limit of the new rational function: \(\lim_{x \to \infty} \frac{15x^2 - 2}{21x^2} = \lim_{x \to \infty} \frac{15x^2}{21x^2} - \frac{2}{21x^2}\).

Step 4: Simplify the Limit

Since \(\frac{15x^2}{21x^2} \rightarrow \frac{15}{21} = \frac{5}{7}\) as \(x \rightarrow \infty\), and \(\frac{2}{21x^2} \rightarrow 0\), the entire expression \(\lim_{x \to \infty} \left( \frac{5}{7} - \frac{2}{21x^2} \right)\) simplifies to \(\frac{5}{7}\).

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form is a situation where direct substitution in a mathematical expression does not initially reveal the actual limit or behavior of the expression. These forms are typically encountered when evaluating limits, especially if both the numerator and denominator of a fraction approach infinity.

The most common indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( \infty - \infty \)
• \( 0 \cdot \infty \)
• \( \infty^0 \)
• \( 1^\infty \)
• \( 0^0 \)

These forms do not straightforwardly convey a real number, requiring further manipulation or advanced calculus techniques such as l'Hopital's rule to resolve. In our exercise, we identify that \( \frac{\infty}{\infty} \) appears when both the numerator and denominator approach infinity as \(x\) grows larger. This confirms the need for l'Hopital's rule to find the limit.

Limits

The concept of limits is a cornerstone in calculus, representing the value that a function or sequence "approaches" as the input approaches some value. Limits are essential for defining derivatives, integrals, and continuity, but they can also tell us a lot about the behavior of functions as they tend toward infinity or any specific point.

For example, in our exercise, as \(x\) approaches infinity, we seek to find out what happens to the expression \(\frac{5x^3 - 2x}{7x^3 + 3}\). By identifying it as an indeterminate form \(\frac{\infty}{\infty}\), we know that directly substituting infinity does not help; instead, we apply calculus techniques like l'Hopital's rule to explore further.

When properly evaluated using l'Hopital's rule, we can redefine the problem in derivative terms: \(\lim_{x \to \infty} \frac{15x^2 - 2}{21x^2}\). Then, by simplifying the expression and logically canceling terms, the effective limit equates as solutions direct towards \(\frac{5}{7}\).

Calculus Concepts

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Its foundation lies within differential calculus and integral calculus, helping describe changes and areas, respectively.

Differential calculus, as applied in l'Hopital’s rule, plays a crucial role in solving limits involving indeterminate forms. By differentiating both the numerator and denominator until terms are manageable, we transform complex problems into simpler forms. This permits a more facile calculation of limits without direct substitution.

Incorporating calculus concepts, like differentiation and limits, allows us to handle problematic expressions seamlessly. In our specific exercise, recognizing polynomial dominance and canceling out terms led us to the limit solution smoothly. By comprehending these interlinked concepts, such as continuity, derivative functions, and polynomial behavior, students can tackle wider ranges of issues in calculus, enriching their problem-solving toolkit.