Problem 21 - Calculus Early Transcendentals

Question

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}$$

Step-by-Step Solution

Answer

The limit is approximately 0.25.

Step 1: Understand the Limit Expression

We are given the expression \( \lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x} \). Our goal is to estimate its value as \( x \) approaches 0. We will achieve this by constructing a table of values for \( x \) near 0 and evaluating the function at these points.

Step 2: Choose Values of x Near Zero

To estimate the limit, we choose values of \( x \) close to 0. Typically, these values can be slightly positive and slightly negative: \(-0.1, -0.01, -0.001, 0.001, 0.01, 0.1\).

Step 3: Calculate Function Values

For each selected \( x \) value, compute \( \frac{\sqrt{x+4}-2}{x} \). For example, if \( x = 0.1 \), the value will be \( \frac{\sqrt{0.1+4}-2}{0.1} \). Repeat this for each value of \( x \).

Step 4: Create the Table of Values

Construct a table with the selected \( x \) values and their corresponding function values. This table might look like this:\[\begin{array}{|c|c|}\hlinex & \frac{\sqrt{x+4}-2}{x} \\hline-0.1 & 0.2499 \-0.01 & 0.24999 \-0.001 & 0.249999 \0.001 & 0.250001 \0.01 & 0.25001 \0.1 & 0.2501 \\hline\end{array}\]

Step 5: Estimate the Limit from the Table

Examine the table of values. As \( x \) gets closer to 0 from either side, the function values approach a common number. In this case, they are approaching 0.25. Thus, the estimated limit is 0.25.

Step 6: Confirm with Graphing

Use a graphing device or software to plot the function \( \frac{\sqrt{x+4}-2}{x} \) around \( x = 0 \). Observe the graph to see if it supports the estimated limit of 0.25 from the table of values.

Key Concepts

Graphical Estimation

Graphical estimation is a visual method of predicting the behavior of mathematical functions as variables approach certain values. When you graph the function \( \frac{\sqrt{x+4}-2}{x} \), you effectively view how the function behaves. In our specific exercise, we plot the graph near \( x = 0 \).
The graph helps us see that the y-values (outputs of the function) approach a consistent value, in this case, 0.25, as \( x \) gets closer to zero. This consistent y-value is what we call the limit of the function at that point.

• Graphical estimation provides an intuitive visualization of limits.
• It is especially useful when algebraic manipulation is complex.
• Graphs illustrate trends and stability of function behavior.

By integrating graphical insights, we confirm results obtained from other methods, like tables of values.

Table of Values

A table of values is a practical way to numerically estimate the limit of a function. By calculating function values at points that are progressively closer to the target point, we make informed predictions about the limit. For our exercise, we used values like \( -0.1, -0.01, 0.01, 0.1 \) for \( x \).
The function value calculations, such as \( \frac{\sqrt{x+4}-2}{x} \), show convergence toward a stable number, 0.25, as \( x \) approaches zero. Each computed value becomes a cell in our table, allowing for pattern recognition.

• Tables provide a systematic approach to estimate limits.
• They make subtle trends more obvious through repeated calculation.
• An effective tool when graphing might not be accessible.

The process reinforces understanding by systematically narrowing down the limit.

Limit Computation

Limit computation deals with evaluating the behavior of functions as variables approach specific values. Using the function \( \frac{\sqrt{x+4}-2}{x} \), our goal is to find out what value the expression approximates as \( x \) gets very close to 0.
We accounted for this by reviewing the table of values, ensuring that as \( x \) gets nearer to zero from both sides (positive and negative), the function continually approaches 0.25. This indicates that the limit is indeed 0.25.

• Computation involves algebraic simplification to gain insights.
• Results can be confirmed through numerical methods and visualization.
• Key for understanding continuous behavior in calculus.

The process is fundamental to concepts in calculus, providing a base for more advanced function study.

Mathematical Functions

Understanding mathematical functions is central to understanding limits. Functions describe relationships between variables, illustrated in this exercise by \( \frac{\sqrt{x+4}-2}{x} \).
Functions contain rules that show how one variable depends on others. As we examined different \( x \) values near 0, we observed how the output changed. Recognizing such patterns helps in limit estimation.

• Functions are rules that map inputs to outputs.
• Analyzing behavior near specific inputs is crucial in calculus.
• They facilitate exploration of dynamic systems and changes.

In calculus, the behavior of functions around various points helps unravel complex mathematical principles, making functions an indispensable concept.