Problem 7 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( fty \).

Step 1: Identify Leading Terms

The given expression is \( \lim_{x \rightarrow \infty} \frac{x^2 - 2}{2x + 1} \). To evaluate this limit, first identify the term with the highest degree in both the numerator and the denominator. In the numerator \( x^2 \) is the leading term and in the denominator \( 2x \) is the leading term.

Step 2: Simplify by Dividing by the Leading Term

Divide every term in both the numerator and the denominator by \( x^2 \) which is the highest power of \( x \) in the numerator. This gives us \( \frac{x^2/x^2 - 2/x^2}{2x/x^2 + 1/x^2} = \frac{1 - 2/x^2}{2/x + 1/x^2} \).

Step 3: Evaluate the Limit as x Approaches Infinity

As \( x \rightarrow \infty \), the terms \( \frac{2}{x^2} \) and \( \frac{1}{x^2} \) in the numerator, and \( \frac{2}{x} \) and \( \frac{1}{x^2} \) in the denominator tend to 0. Thus, the expression simplifies to \( \frac{1 - 0}{0 + 0} \).

Step 4: Resolve the Simplified Limit

After simplification, the limit approaches \( \frac{1}{0} \), suggesting the result tends to infinity. Therefore, the original limit is \( fty \).

Key Concepts

Evaluating Limits

In calculus, evaluating limits is a fundamental concept. It involves finding the value that a function gets closer to as the input approaches a specific point. In our example, we are looking at the limit as \(x\) approaches infinity, written as \(\lim_{x \rightarrow \infty}\). This means we are interested in the behavior of the function when \(x\) becomes very large.
The process of evaluating limits often requires simplifying the expression. Here are some helpful steps used in evaluating limits:

• Identify the dominant terms: These are the terms with the highest power in both the numerator and denominator, as they predominantly affect the outcome of the limit.
• Simplify the expression: Divide each term by the leading term from the numerator to make solving easier.
• Consider the behavior as \(x\) grows: Terms involving division by \(x\) tend to zero as \(x\) becomes very large. Therefore, they could be removed from calculations when evaluating limits at infinity.

These steps help in simplifying complex expressions and making the behavior of functions at infinity more apparent.

Rational Functions

Rational functions are special types of functions represented by the ratio of two polynomials. In the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, rational functions can demonstrate various behaviors based on their degree.
In our example, \(f(x) = \frac{x^2 - 2}{2x + 1}\), the numerator has a polynomial of degree 2, while the denominator has a degree of 1. The degree of a polynomial is determined by the highest power of \(x\) it contains. For rational functions, the degree of the numerator and denominator plays a crucial role in determining the limit behavior at infinity:

• If the degree of the numerator is greater than the degree of the denominator, the limit tends to infinity or negative infinity.
• If the degree of the numerator equals the degree of the denominator, the limit is the ratio of the leading coefficients.
• If the degree of the numerator is less than the degree of the denominator, the limit tends to zero.

Understanding these principles helps in evaluating and predicting the behavior of rational functions as \(x\) approaches infinity.

Infinite Limits

Infinite limits occur when the value of a function grows larger or smaller without bound as \(x\) approaches a certain point, often leading to infinity \(\infty\). When analyzing functions, particularly rational functions, these situations frequently arise.
In the original exercise, we simplify the expression \(\frac{x^2 - 2}{2x + 1}\) as \(x\) approaches infinity. After simplification and eliminating zero-influenced terms in the fraction, we find that the limit of the function approaches \(\infty\). This is an example of an infinite limit, where the function does not settle around a particular finite number.

• Infinite limits inform us about the long-term behavior of a function, often exhibiting that a function increases indefinitely.
• An important aspect is understanding whether this goes towards positive or negative infinity, determined by the sign of the leading terms.

Recognizing infinite limits is essential, as they help predict and understand the asymptotic behavior of functions, especially in calculus and real-world applications.