Problem 12 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{2 x+x^{2}}{3 x+1} $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Highest Power of x

In the given expression \( \frac{2x + x^2}{3x + 1} \), identify the highest power of \( x \) in both the numerator and the denominator. Here, the highest power in the numerator is \( x^2 \), and in the denominator, it is \( x \).

Step 2: Divide Every Term by the Highest Power in the Denominator

Divide each term in the numerator and denominator by \( x \), the highest power found in the denominator. The expression becomes \( \lim_{x \to -\infty} \frac{\frac{2x}{x} + \frac{x^2}{x}}{\frac{3x}{x} + \frac{1}{x}} \).

Step 3: Simplify the Expression

Simplifying, we have: \( \lim_{x \to -\infty} \frac{2 + x}{3 + \frac{1}{x}} \). Now, consider the behavior of each term as \( x \to -\infty \). The term \( \frac{1}{x} \) approaches 0.

Step 4: Evaluate the Limit

Substituting the values as \( x \to -\infty \), in the simplified limit, the expression becomes \( \frac{2 + (-\infty)}{3} = \frac{-\infty}{3} \). This evaluates to \( -\infty \).

Key Concepts

Understanding the Highest Power of x

In problems involving limits, especially with rational functions, identifying the highest power of \( x \) is crucial. Let’s look at the example of \( \frac{2x + x^2}{3x + 1} \). In this expression, we must recognize which terms dominate the behavior of the function as \( x \) becomes very large or very small. Here’s the key idea:

• The highest power of \( x \) in the numerator is \( x^2 \).
• The highest power of \( x \) in the denominator is \( x \).

Why does this matter? Because as \( x \) approaches infinity or negative infinity, terms with lower powers of \( x \) grow insignificant compared to those with the highest power. This helps simplify the process of finding limits by focusing on dominant terms in both the numerator and the denominator.

Rational Functions and Their Limits

Rational functions consist of a polynomial in the numerator and another in the denominator. Understanding their limit behavior is about simplifying these functions intelligently.
To identify a rational function's limit as \( x \) approaches a very large number (positive or negative infinity), we follow these steps:

• Determine the highest power of \( x \) in both the numerator and denominator.
• Divide every term by the highest power found in the denominator.

By simplifying the expression \( \frac{2x + x^2}{3x + 1} \) as \( x \to -\infty \), the expression becomes \( \lim_{x \to -\infty} \frac{2 + x}{3 + \frac{1}{x}} \). Here, you can see that the expression's limit largely depends on the highest power terms. This is a streamlined approach, making complex calculations more manageable.

Behavior as x Approaches Infinity

Exploring the behavior as \( x \) approaches infinity or negative infinity is all about knowing which terms affect the limit’s value. In our expression \( \lim_{x \to -\infty} \frac{2 + x}{3 + \frac{1}{x}} \), each term's behavior matters:

• The term \( x \) in the numerator becomes dominant as it trends towards \(-\infty\), overpowering the constant \( 2 \).
• In the denominator, the term \( \frac{1}{x} \) approaches \(0\), making it negligent.

Given these behaviors, when you substitute the infinite values, \( \frac{2 + (-\infty)}{3} \) results in \( \frac{-\infty}{3} \), simplifying to \(-\infty\). This illustrates how larger powers drive the behavior, influencing the resulting limit significantly.