Problem 29 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 0} \dfrac{\dfrac{1}{x+4}-1}{x}$$

Step-by-Step Solution

Answer

The limit as x approaches 0 is 3/4.

Step 1: Simplify the Complex Fraction

Rewrite the given complex fraction, our first step will be to find a common denominator so that we can subtract the two fractions in the numerator. The common denominator will be \(x+4\). So the equation becomes \( \lim_{x \to 0} \dfrac{(x+4-1)}{x(x+4)} = \lim_{x \to 0} \dfrac{x+3}{x(x+4)}\)

Step 2: Factor and Cancel

Now we will factor and cancel any common factors from the numerator and denominator. Since \(x\) is a common factor, cancel it out and we get: \( \lim_{x \to 0} \dfrac{(x+3)}{(x+4)} \)

Step 3: Evaluate the Limit

Substitute \(x=0\) into the function and solve: \((0+3)/(0+4) = 3/4\)

Step 4: Graphical Verification

This step will be for verifying the result using a graphing utility. It might show as a hole in the graph at \(x=0\) since original function was not defined there, but it still tends to a y-value of 3/4 as x gets closer to 0.

Key Concepts

Understanding Complex Fractions

Complex fractions are just fractions where the numerator, the denominator, or both contain fractions themselves. To simplify them, you often need to find a common denominator. For example, if you have a complex fraction like \( \frac{\frac{1}{x+4} - 1}{x} \), the denominator for the terms in the numerator is \(x+4\). By using it, you combine the fractions in the numerator, simplifying the whole expression. The main goal is to rewrite the fraction in a form that makes it easier to evaluate, especially at specific points of interest, such as when taking limits.

Using Graphing Utilities for Verification

Graphing utilities are powerful tools that allow you to visualize mathematical expressions. They provide a graphical perspective which can make understanding limits and functions much clearer. When you graph the function \( \frac{x+3}{x(x+4)} \), initially derived from our complex fraction, a graphing utility can show you the behavior near \(x=0\). Even when the fraction is undefined at \(x=0\), due to division by zero, graphing utilities can illustrate the limit by showing how the function behaves as x approaches that point. Look for it to get closer and closer to a specific value, often indicating the limit.

Connecting Limits with Calculus Concepts

Calculus revolves around concepts like limits, which form the foundation of derivatives and integrals. Limits help us understand the behavior of functions as they approach certain points, even if the function doesn't exist at that point. In the exercise, the limit \( \lim_{x \to 0} \frac{x+3}{x(x+4)} \) was evaluated to show how it approaches \( \frac{3}{4} \) as \(x\) approaches 0. This concept of approaching a value sets the stage for further calculus work, such as finding instantaneous rates of change or calculating the area under a curve. By grasping limits, you're building the blocks for deeper calculus studies.