Problem 27 - Precalculus Mathematics for Calculus

Question

(a) Estimate the value of $$\lim _{x \rightarrow 0} \frac{x}{\sqrt{1+3 x}-1}$$ by graphing the function \(f(x)=x /(\sqrt{1+3 x}-1)\) (b) Make a table of values of \(f(x)\) for \(x\) close to \(0,\) and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

Step-by-Step Solution

Answer

The estimated and verified limit is \( \frac{2}{3} \).

Step 1: Understanding the Function

We begin by analyzing the function to estimate the limit: \( f(x) = \frac{x}{\sqrt{1+3x} - 1} \). This function is undefined at \( x = 0 \) as it leads to division by zero. Therefore, we'll seek ways to simplify or approach \( x = 0 \) using graphing and tabulation to estimate the limit.

Step 2: Graphing the Function

We graph the function \( f(x) = \frac{x}{\sqrt{1+3x} - 1} \) around \( x = 0 \). Close examination of the graph as \( x \) approaches \( 0 \) from both the positive and negative sides suggests that the function stabilizes around a certain value. The graph shows that the function appears to approach the value \( \frac{2}{3} \).

Step 3: Creating a Table of Values

We create a table with values of \( f(x) \) by choosing small positive and negative values of \( x \) around \( 0 \). For each chosen \( x \), compute \( f(x) \) :- \( x = 0.1 \), \( f(x) \approx 0.6667 \)- \( x = 0.01 \), \( f(x) \approx 0.6667 \)- \( x = -0.01 \), \( f(x) \approx 0.6667 \)- \( x = -0.1 \), \( f(x) \approx 0.6667 \)From these values, we deduce that \( \lim_{x \to 0} f(x) = \frac{2}{3} \).

Step 4: Proving the Limit using Rationalization

To confirm via Limit Laws, we rationalize the expression \( \frac{x}{\sqrt{1+3x} - 1} \). Multiply numerator and denominator by the conjugate: \( \frac{x(\sqrt{1+3x} + 1)}{(\sqrt{1+3x}-1)(\sqrt{1+3x}+1)} = \frac{x(\sqrt{1+3x} + 1)}{3x} \). Simplify to get \( \frac{\sqrt{1+3x} + 1}{3} \). As \( x \rightarrow 0 \), \( \sqrt{1+3x} \rightarrow 1 \), thus the expression becomes \( \frac{2}{3} \).

Step 5: Conclusion

We conclude that both our estimation methods (graphing and tabulation) and analytical proof using Limit Laws agree that the limit is \( \frac{2}{3} \). Therefore, the solution is confirmed, and the limit is indeed \( \frac{2}{3} \).

Key Concepts

Limit Laws

In calculus, limit laws are essential tools that help us evaluate limits of functions more easily and systematically. These laws provide algebraic shortcuts to simplify complex expressions as a variable approaches a specific value. For the function given in the exercise, the limit laws facilitate proving the limit is \( \frac{2}{3} \) at \( x = 0 \).
When dealing with limits, key laws include:

• The sum law: The limit of a sum is equal to the sum of the limits.
• The product law: The limit of a product is equal to the product of the limits.
• The quotient law: The limit of a quotient is equal to the quotient of the limits, as long as the denominator isn't zero.

In our problem, rationalization was key in using these laws effectively. By rationalizing, we reform the expression in such a way as to remove the potential for division by zero, leading to a clearer application of the quotient law and showing the result \( \frac{2}{3} \). Understanding these laws assists in analyzing functions and simplifies finding their limits.

Rationalization

Rationalization is a technique used to simplify expressions by removing radicals from the denominator. This is especially useful in limit problems where direct substitution leads to an undefined form such as \( \frac{0}{0} \). By rationalizing, we can restructure the expression to avoid such issues, enabling a clear path to evaluate the limit.
In the given exercise, the function \( f(x) = \frac{x}{\sqrt{1+3x} - 1} \) is initially undefined at \( x = 0 \). By multiplying the numerator and the denominator by the conjugate \( \sqrt{1+3x} + 1 \), the expression transforms into a more manageable form:

• The denominator \( (\sqrt{1+3x} - 1)(\sqrt{1+3x} + 1) \) simplifies to \( 3x \).
• The entire expression simplifies to \( \frac{\sqrt{1+3x} + 1}{3} \).

After rationalization, substituting \( x = 0 \) is straightforward, resulting in the limit \( \frac{2}{3} \). This process not only confirms the limit but also demonstrates rationalization as a potent tool in transforming complex expressions into simpler forms for analysis.

Graphing Functions

Graphing functions is an invaluable method for visually exploring the behavior of mathematical expressions. By plotting a graph of a function, we gain insight into its behavior around specific points—like where the limit is to be determined—as well as a general understanding of the function's trend.
In this exercise, graphing \( f(x) = \frac{x}{\sqrt{1+3x} - 1} \) around \( x = 0 \) shows how the function behaves as it approaches the point. Observing the plot, the graph appears to converge towards the value \( \frac{2}{3} \). This visual estimate provided a guideline for detailed analytical verification and can often confirm our numerical or algebraic guess about a limit.
Graphing offers an initial checkpoint when analyzing limits, as it presents a qualitative sense of the function's behavior. This becomes especially useful in validating results obtained through limit laws and rationalization.