Problem 26 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to \infty}\frac{x + 3x^2}{4x - 1} \)

Step-by-Step Solution

Answer

The limit does not exist.

Step 1: Identify the Highest Power

Examine the limit expression: \( \lim_{x \to \infty}\frac{x + 3x^2}{4x - 1} \). Notice that the highest power of \( x \) in the numerator is \( x^2 \) and in the denominator is \( x \). Thus, the dominant term in the numerator is \( 3x^2 \) and in the denominator is \( x \).

Step 2: Simplify by Dividing Through by \( x^2 \)

To simplify, divide each term in the numerator and the denominator by \( x^2 \). This gives: \( \frac{\frac{x}{x^2} + \frac{3x^2}{x^2}}{\frac{4x}{x^2} - \frac{1}{x^2}} = \frac{\frac{1}{x} + 3}{\frac{4}{x} - \frac{1}{x^2}} \).

Step 3: Evaluate the Limit

Evaluate each term as \( x \) approaches infinity: \( \frac{1}{x} \to 0 \), \( \frac{4}{x} \to 0 \), and \( \frac{1}{x^2} \to 0 \). Substitute these results back into the expression to obtain \( \frac{0 + 3}{0 - 0} = \frac{3}{0} \).

Step 4: Determine the Behavior of the Limit

The expression \( \frac{3}{0} \) shows that the numerator has a non-zero value while the denominator approaches zero, which implies that the limit cannot stabilize to a finite number. Therefore, the limit does not exist as \( x \to \infty \).

Key Concepts

Limits at Infinity

When evaluating limits at infinity, we try to understand how a function behaves as the value of its variable grows larger without bound. In the exercise provided, we deal with the expression \( \lim_{x \to \infty}\frac{x + 3x^2}{4x - 1} \). This asks us to find out what happens to the function as \( x \) becomes infinitely large.
Limits at infinity are crucial in calculus because they help us comprehend the end behavior of functions and their asymptotic behavior.

• If a function approaches a specific value as \( x \to \infty \), that value is the limit at infinity.
• Sometimes, like in our case study, a limit doesn't exist if the function doesn't stabilize to a finite value.

Understanding this helps us graph functions and analyze trends in data.

Dominant Term Analysis

Dominant term analysis simplifies evaluating limits by focusing on the most significant contributors to the output as \( x \to \infty \). In any algebraic expression, the terms with the highest power of \( x \) generally dominate the behavior as \( x \) grows.
For the exercise example, the dominant term in the numerator is \( 3x^2 \), and the dominant one in the denominator is \( 4x \). Why are these dominant? As the variable \( x \) becomes very large, these terms grow much faster than the others, effectively "drowning out" lesser terms. By identifying and focusing on these dominant terms, we can simplify our calculations greatly.
Simplifying the expression involves dividing each term by \( x^2 \), the highest power present in the numerator, leading us to calculate limits efficiently.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. The exercise is an example of a rational function: \( \frac{x + 3x^2}{4x - 1} \). These functions are essential in calculus due to their versatile applicability across various fields and their interesting properties.
When handling limits, a crucial step involves understanding how the degrees of these polynomials affect the limit's behavior.

• If the degree in the numerator is greater than in the denominator, often the limit at infinity doesn't exist (as in this exercise).
• If the degrees are equal, the limit is typically the ratio of the leading coefficients.
• If the degree in the numerator is less, the limit often approaches zero.

Understanding rational functions helps chart their behavior, identify asymptotes, and predict trends.

Behavior of Functions as x Approaches Infinity

Analyzing the behavior of functions as \( x \to \infty \) allows us to predict and understand how a function acts without graphing it. In our problem, finding \( \lim_{x \to \infty}\frac{x + 3x^2}{4x - 1} \) shows us that the function tries to reach a state but is disrupted by the denominator approaching zero, causing the limit not to exist.
When considering such behavior, it is useful to:

• Identify dominant terms which will govern the outcome as \( x \) becomes large.
• Divide through by the highest power of \( x \) in the dominant term to simplify the analysis.
• Check if each component's limit can simplify the expression simplistically.

Doing so helps in evaluating not just limits at infinity but also gaining insight into a function's asymptotic behavior, showing us that sometimes functions don't settle into a steady state as \( x \) increases.