Problem 48 - Calculus for Biology and Medicine

Question

In Problems 39-56, 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

The expression given is \( \lim_{x \rightarrow -2} \frac{1+x}{1-x} \). We need to evaluate this limit as \( x \) approaches \(-2\).

Step 2: Apply Limit to Numerator and Denominator Separately

According to the limit laws, we first consider the limit of the numerator \(1+x\) and the limit of the denominator \(1-x\) separately as \(x\) approaches \(-2\).

Step 3: Evaluate the Limit of the Numerator

Calculate the limit of the numerator expression: \( \lim_{x \rightarrow -2} (1+x) = 1 + (-2) = -1 \).

Step 4: Evaluate the Limit of the Denominator

Calculate the limit of the denominator expression: \( \lim_{x \rightarrow -2} (1-x) = 1 - (-2) = 1 + 2 = 3 \).

Step 5: Apply the Quotient Rule

Using the quotient rule for limits, compute the limit of the entire expression: \( \lim_{x \rightarrow -2} \frac{1+x}{1-x} = \frac{-1}{3} \).

Step 6: Confirm Result is Well-Defined

Ensure the denominator does not approach zero, which it doesn't in this case since \( 1-x \) approaches \(3\). Therefore, the limit exists and is \(-\frac{1}{3}\).

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental principle in calculus for handling limits of fractions. When you're working with the limit of a fraction, like the one we have \(\lim_{x \rightarrow -2} \frac{1+x}{1-x}\), the quotient rule tells us how to break it down. The rule states that if both the numerator and denominator have defined limits as they approach a certain point, and the limit of the denominator is not zero, then we can find the overall limit of the fraction by:

• Taking the limit of the numerator.
• Taking the limit of the denominator.
• Dividing these two results.

In our original exercise, we applied this rule and found \(\lim_{x \rightarrow -2} (1+x) = -1\) and \(\lim_{x \rightarrow -2} (1-x) = 3\). Therefore, \[\lim_{x \rightarrow -2} \frac{1+x}{1-x} = \frac{-1}{3}.\] For applications, just remember that the quotient rule only works if the denominator doesn't turn into zero, otherwise calculations can lead to undefined results.

Limit Evaluation

Limit evaluation builds the bridge between variable unknowns and fixed results. It's like predicting the behavior of a mathematical function as it approaches a specific point. In our exercise, we were asked to evaluate \(\lim_{x \rightarrow -2} \frac{1+x}{1-x}\). To do this, we broke the problem into smaller, more manageable parts.First, we evaluated the numerator and the denominator separately. This is where limit laws come in handy; they help simplify such complex expressions. As \(x\) approaches \(-2\), the numerator \(1+x\) becomes \(-1\), and the denominator \(1-x\) becomes \(3\). Together, these steps show how we predictably transformed the expression into a simple fraction result using basic arithmetic.

Applied Calculus

Applied Calculus showcases how calculus tools are used to solve real-world problems and more complex mathematical equations. In the context of our limit example, it gives us a method to compute exact numbers for seemingly ambiguous or undefined expressions. It allows us to:

• Systematically deconstruct equations.
• Apply rules like the product, sum, and quotient rules where necessary.
• Confirm our results by checking our assumptions and calculations.

Even though at a first glance a limit problem like our example might look straightforward, the procedure is crucial in more complicated scenarios. With these skills, one can tackle dynamics in physics, optimizations in economics, and growth projections in biology. Calculus lays the backbone for interpretations in several fields, showing why it's often called the 'language of sciences.'