Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0^{-}} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline x & -0.5 & -0.1 & -0.01 & -0.001 & 0 \\ \hline f(x) & & & & & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 0 from the left is '0'. This can be confirmed by graphing the function.

Step 1: Simplify function

The given function is \( \frac{\frac{1}{x+4}-\frac{1}{4}}{x} \). Simplify the numerator using the common denominator technique, which results in \( \frac{4 - (x+4)}{4x(x+4)} \) which simplifies further to \( \frac{-x}{4x(x+4)} \) or \( \frac{-1}{4(x+4)} \).

Step 2: Compute function values

Using this simplified function, substitute the given 'x' values into the function: For \(x = -0.5\), the function becomes -0.13333, for \(x = -0.1\), the function is -0.02439, for \(x = -0.01\), the function becomes -0.00244 and for \(x = -0.001\), function is -0.000244.

Step 3: Estimate the limit

After seeing the function values at the proximities of '0', it can be seen that as 'x' approaches 0 from the left, the function is approaching '0'. Therefore, the limit of the function as x approaches 0 from the left, \( \lim _{x \rightarrow 0^{-}} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} \), is '0'.

Step 4: Graph confirmation

Graph the function to confirm your results. The graph should show the function approaching '0' as 'x' tends to '0' from the left. Any graphing utility can be used to confirm this.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial to the study of calculus. It involves determining the value that a function approaches as the input (or 'x' value) gets closer to some point. For instance, in our exercise, we want to know what value \( f(x) \) gets close to as \(x\) approaches 0 from the left (notated as \(x \rightarrow 0^{-}\)). The beauty of limits is that it doesn't matter what the function's value is exactly at '0', just what value it is approaching.

To estimate the limit, you can calculate the function's value at points getting ever closer to '0' from the negative direction. If the results of \( f(x) \) are becoming increasingly insignificant towards a specific number, that number is where the function is heading — its limit. In our textbook's exercise, it turns out this limit is '0'.

Graphing Utility

A graphing utility is an invaluable tool for visualizing functions and confirming theoretical outcomes like limits. You can input your function's equation and observe the curve or particular points on a graph. When we talk about estimating the limit, as \(x\) approaches a certain value, graphing that function can give us a clear picture of where the values are headed.

In our example, graphing \(f(x)\) as \(x\) approaches zero from the left-hand side should display the curve getting closer and closer to the '0'. This visualization serves as confirmation for the numerical estimation we did earlier in the steps, and with technology so readily available, this method is both quick and precise.

Algebraic Simplification

Algebraic simplification is all about making complex expressions easier to work with. It's like untangling a mess of string into a straight line. By using algebraic techniques, such as combining like terms or factoring, you can often rewrite a function in a more manageable form.

In our exercise, we see algebraic simplification at play when we rework the complex fraction within the function into a simpler fraction. This is done by finding a common denominator, and then subtracting the two fractions in the numerator before ultimately cancelling terms where possible. Simplifying the given function makes it easier to substitute values for \(x\) and to understand the behavior of the function as \(x\) approaches the limit.

Common Denominator Technique

When faced with an algebraic expression that involves multiple fractions, like in our given problem, the common denominator technique becomes your best friend. This method involves finding a common denominator for all fractions involved, which allows you to combine them into a single fraction. It's just like finding a common ground so everyone can join the same conversation.

We put this technique into action by combining \(\frac{1}{x+4}\) and \(\frac{1}{4}\) into one fraction. The common denominator here is \(4(x+4)\), and once we apply this, we can execute the subtraction which leads to further simplification. It's a powerful skill that makes limits and other calculus problems much easier to deal with, and it's something you'll want to be proficient in as you study more advanced mathematics.