Problem 31 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The limit as \(x\) approaches 0 equals to \(-1/16\).

Step 1: Simplify the expression

We can first simplify the original expression by finding a common denominator for the fractions in the numerator. Thus, the expression becomes: \[\lim_{x \to 0} \dfrac{4-(x+4)}{4(x+4)x}\] which then simplifies to \[\lim_{x \to 0} \dfrac{-x}{4x(x+4)}.\] After cancelling \(x\) in the numerator and denominator, the expression is \[\lim_{x \to 0} \dfrac{-1}{4(x+4)}.\]

Step 2: Evaluate the limit

Substitute \(x=0\) into the simplified expression: \[\lim_{x \to 0} \dfrac{-1}{4(x+4)} = \dfrac{-1}{4(0+4)} = -\dfrac{1}{16}.\]

Step 3: Verify with a graphing utility

A graphing utility can be used to graph the function and verify that the limit as \(x\) approaches 0 equals to \(-1/16\).

Key Concepts

Graphing Utility

Visualizing mathematical expressions is a fantastic way to gain deeper insights. Graphing utilities are tools that allow us to see the behavior of a function as the variable approaches a certain value. For the above problem, we can use a graphing utility to plot the function \( f(x) = \frac{\frac{1}{x+4}-\frac{1}{4}}{x} \).

• Start by entering the function into the graphing utility.
• Set the graph to visualize the area around \(x = 0\).
• Observe how the graph behaves as \(x\) approaches 0.
• Note that while the function is undefined at \(x = 0\) due to division by zero, the behavior around this point gives an indication of the limit.

Looking at the graph, you'll notice the curve approaches a specific horizontal line as \(x\) nears zero. This line represents the limit value which, as you verified, is \(-\frac{1}{16}\). This graphical approach not only supports the algebraic solution but provides an intuitive understanding of the limit.

Algebraic Simplification

Simplifying expressions is critical to understanding limits, especially ones involving complex fractions. In this exercise, simplifying the expression before calculating the limit made everything much easier.
Here's how you can simplify it:

• Start by rewriting the numerator: \( \frac{1}{x+4} - \frac{1}{4} \) can be combined into a single fraction by finding a common denominator.
• The common denominator for \(x+4\) and \(4\) is \(4(x+4)\), allowing us to rewrite the numerator as \( \frac{4 - (x+4)}{4(x+4)} \).
• This simplifies to \( \frac{-x}{4(x+4)} \).
• Cancel the \(x\) in the numerator and denominator, which reduces to \( \frac{-1}{4(x+4)} \).

After simplifying, evaluating the limit becomes straightforward. Indeed, simplification paves the way to solving the problem without stumbling on undefined terms.

Limit Verification

Verifying the result of a limit calculation, especially after simplification, reassures us of the expression's correctness. Once we algebraically determined that \( \lim_{x \to 0} \frac{-1}{4(x+4)} = -\frac{1}{16} \), we still needed to ensure our solution aligns with the actual behavior of the expression.
Let's verify:

• Substitute \(x = 0\) into the simplified expression. Ensure no division by zero occurs with the cancellation previously done.
• Calculate the expression: \( \frac{-1}{4(0+4)} = \frac{-1}{16} \).
• Using a graphing utility, check the graph to see if it indeed approaches \(-\frac{1}{16}\) as \(x\) tends towards 0.

Confirmation from both algebraic and graphical methods strengthens the correctness of the solution. This dual approach enhances understanding, showcasing how algebra and technology together clarify complex mathematical concepts.