Question

In Exercises \(1-8,\) complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 3} \frac{[1 /(x+1)]-(1 / 4)}{x-3}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 2.9 & 2.99 & 2.999 & 3.001 & 3.01 & 3.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} \end{aligned} $$

Short answer

The actual limit value would depend upon the calculated function values. However, the limit of this function as \(x\) approaches 3 will be the number that the function values converge to as \(x\) approaches 3 from both left and right. The graph of the function should also confirm this limit.

Step-by-step solution

Calculate the Function Values

For each of the \(x\) values in the table (2.9, 2.99, 2.999, 3.001, 3.01, 3.1), compute the value of the function \(f(x)\) = \(\frac{1}{x+1} - \frac{1}{4}\). This can be done by substituting each \(x\) value into the function and solving. Use a calculator to make these calculations as they involve fractional numbers.

Estimate the Limit

Using the \(f(x)\) values obtained in Step 1, make an approximation of the limit of the function as \(x\) approaches 3. The closer \(x\) is to 3, the closer to the actual limit the function value should be. If the function values are converging to a specific number, that would be the estimate for the limit.

Graph the Function

Plot the function \(f(x)\) = \(\frac{1}{x+1} - \frac{1}{4}\) using a graphing utility to confirm the estimated limit. Look for the value of the function as \(x\) gets very close to 3. The function value at this point should be around the same as the calculated limit from Step 2.

Key concepts

Limits in Calculus

Understanding limits in calculus is a fundamental aspect that helps in studying the behavior of functions as inputs approach a certain value. When we talk about the limit of a function as the input approaches a particular value, we are essentially describing what value the output of the function gets closer to, even if the function is not defined at that input value.

For example, in the provided exercise, we are interested in finding \(lim _{x \rightarrow 3} \frac{[1 /(x+1)]-(1 / 4)}{x-3}\). The notation \(x \rightarrow 3\) indicates that we are looking at what happens to our function as \(x\) gets closer and closer to 3. We are not concerned with the function's value exactly at 3, especially since it may not even be defined there, but rather the value it approaches as close as we can imagine to 3 from both sides.

The essence of finding limits is central to the study of calculus as it leads to the formal definition of the derivative, informs the continuity of functions, and is used in evaluating integrals, which are all key concepts and techniques in calculus.

Graphing Functions

Graphing functions is a visual representation of equations on a coordinate plane and is an indispensable tool for understanding the behavior of functions. It allows us to see where a function increases or decreases, where it has maxima and minima, and crucially for this exercise, where it is approaching as the input nears some value.

Using a graphing utility or calculator as suggested in Step 3 of the solution can be particularly enlightening because it shows us visually what's happening as x approaches 3. It aids us in developing intuition about the function's behavior and, in many cases, will confirm the numerical findings we get from our limit estimation strategies. Essentially, graphically analyzing functions complements the numerical techniques by offering a visual corroboration of what numbers imply.

Limit Estimation Strategy

A limit estimation strategy involves coming up with an approximate value for the limit based on available information. In the context of the given exercise, we estimate the limit by calculating the function's value at points increasingly close to 3 from both the left and the right.

Here is where the step-by-step completion of the provided table comes into play. By calculating and tabulating the function's values at 2.9, 2.99, 2.999 (approaching 3 from the left) and at 3.001, 3.01, 3.1 (approaching 3 from the right), we get a series of function values that should become closer to the limit as the x-values get closer to 3. This numerical approach gives us a concrete sequence of numbers to analyze for patterns, making the estimation of the limit a more tangible task.

Calculating Function Values

Calculating function values is the bedrock of effectively understanding any function's behavior. In the exercise, to estimate limits, we first need to evaluate the function at specific points. Calculations should be performed with care to avoid any arithmetic errors, which could lead to incorrect limit estimations.

As shown in Step 1, this involves substituting values that are near our point of interest, here 3, into our function and simplifying. Each chosen value provides a snapshot of the function's behavior near the limit point. The sequence of these snapshots, when examined collectively as we did when we filled in our table, allows us to make a reasoned guess about the function's limit. To reiterate, the calculated function values are a means to end; they are the raw data that we use in crafting our estimate of the limit.