Problem 46 - Calculus for Biology and Medicine

Question

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-2} \frac{1+x}{1-x} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{3}\).

Step 1: Identify the Limit Expression

We begin by examining the function given in the limit: \( \frac{1+x}{1-x} \). We need to find the limit of this function as \( x \) approaches \(-2\).

Step 2: Substitute the Value

The first step in evaluating a limit is to substitute the value \(-2\) into the function: \( \frac{1+(-2)}{1-(-2)} \).

Step 3: Simplify the Numerator

Calculate the numerator by simplifying \( 1 + (-2) \), which results in \(-1\).

Step 4: Simplify the Denominator

Calculate the denominator by simplifying \( 1 - (-2) \), which results in \( 1 + 2 = 3 \).

Step 5: Evaluate the Limit

With the numerator and denominator simplified, we have \( \frac{-1}{3} \). Therefore, the limit evaluates to \( -\frac{1}{3} \).

Key Concepts

Understanding Limit Laws

Limit laws provide the foundational rules for evaluating the limits of different functions. They simplify the process by establishing how limits behave with various mathematical operations. Here are some key limit laws:

• Sum/Difference Law: The limit of a sum or difference is the sum or difference of the limits. \[ \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) \]
• Product Law: The limit of a product is the product of the limits. \[ \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \]
• Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. \[ \lim_{x \to c} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \]
• Constant Law: The limit of a constant is the constant itself. \[ \lim_{x \to c} k = k \]

These laws are crucial in evaluating limits because they allow us to break down complex expressions into manageable parts.

Solving Calculus Problems with Limits

Calculus problems often involve finding limits, which represent the value that a function approaches as the input approaches a certain point. Solving these problems requires a clear understanding of both the function's behavior and the applicable limit laws.

In the given exercise, we deal with a rational function, which is a ratio of two polynomials. To evaluate its limit as \(x\) approaches \(-2\), we can often try direct substitution. If substituting results in a well-defined numerical value with no division by zero or indeterminate form, we have found our limit.

However, if you encounter an indeterminate form such as \(\frac{0}{0}\), additional techniques like factoring, rationalizing, or using L'Hôpital's Rule might be necessary. In our problem, substituting directly yielded a determinate form \(\frac{-1}{3}\), thereby making it straightforward to solve.

Breaking Down Step by Step Solutions

A step-by-step solution helps solidify your understanding by demonstrating exactly how to apply concepts in specific situations. Let's revisit and deepen the steps from the solution:

• Identify the Expression: Recognize the target of the limit operation and the point \(x\) is approaching. This initial step sets the stage for correctly applying the limit laws.
• Substitute the Value: Attempt to plug in the approaching value directly into the function. This straightforward action often gives a quick solution when conditions are right.
• Simplify Numerator and Denominator: Simplifying provides clarity, reducing potential errors in calculation. Here, simplification resulted in fraction \(\frac{-1}{3}\).
• Evaluate the Limit: Finally, interpret the simplified expression to determine the limit. If substitutions lead to a clear and finite result, you have successfully evaluated the limit.

Following a methodical approach not only aids in solving the current problem but also builds confidence for tackling more complex calculus problems.