Problem 5 - 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. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 0 is \( \frac{1}{2} \)

Step 1: Calculate the values for f(x)

First, substitute the given x values into the function \( \frac{\sqrt{x+1}-1}{x} \) and calculate the corresponding values for f(x). Do that for all the listed x values except x = 0, where the denominator would be 0.

Step 2: Use L'Hopital's rule

Since the limit is in an indeterminate form \( \frac{0}{0} \), apply L'Hopital's rule, which states that such limit of a ratio of two functions is equal to the limit of the ratios of their derivatives. That is, \( \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} = \lim _{x \rightarrow 0} \frac{\frac{1}{2\sqrt{x+1}}}{1} \)

Step 3: Find the limit

After applying L'Hopital's rule, calculate the limit as x tends to 0. \( \lim _{x \rightarrow 0} \frac{1}{2\sqrt{x+1}} = \frac{1}{2} \)

Step 4: Graph the Function

Plot the function \( \frac{\sqrt{x+1}-1}{x} \) using a graphing tool and observe the behavior as x tends to 0 to confirm the result.

Key Concepts

L'Hopital's Rule

When you find a limit that results in an indeterminate form such as \( \frac{0}{0} \), L'Hopital's Rule is a very useful tool to evaluate it. By taking the derivatives of the numerator and the denominator separately, you can often simplify the problem. In this exercise, we have the initial function \( \frac{\sqrt{x+1} - 1}{x} \), and at \( x = 0 \), both the numerator and the denominator yield zero. This is where L'Hopital's Rule steps in to help!

• First, differentiate the top: The derivative of \( \sqrt{x+1} - 1 \) is \( \frac{1}{2\sqrt{x+1}} \).
• Then, differentiate the bottom: The derivative of \( x \) is 1.

Applying L'Hopital's Rule, the new expression is \( \lim_{x \to 0} \frac{\frac{1}{2\sqrt{x+1}}}{1} \). This is now straightforward to evaluate as \( x \to 0 \), which gives \( \frac{1}{2} \). L'Hopital's Rule simplifies the process dramatically when facing indeterminate forms.

Indeterminate Forms

When trying to evaluate limits, you might encounter indeterminate forms like \( \frac{0}{0} \) or \( \infty - \infty \). These forms arise when substitution directly into the limit leads to an undefined expression. In this exercise, substituting \( x = 0 \) into \( \frac{\sqrt{x+1} - 1}{x} \) gives \( \frac{0}{0} \), which doesn't provide any real information about the limit.Indeterminate forms signal that we need more sophisticated tools or algebraic techniques to resolve them:

• Factoring and simplifying expressions can sometimes resolve indeterminacies.
• Other times, expanding using series or applying L'Hopital's Rule becomes necessary.

The concept of indeterminate forms is crucial because it identifies situations where direct evaluation fails, pushing us to explore alternative methods like differentiation, which L'Hopital's Rule provides.

Graphing Functions

Graphing a function is like taking a visual approach to understanding its behavior. Observing a graph makes complex mathematical computations tangible and easier to interpret. In this exercise, graphing the function \( \frac{\sqrt{x+1} - 1}{x} \) around \( x = 0 \) helps visually confirm the result obtained analytically.Here’s why graphing is beneficial:

• It provides immediate insight into the limit's behavior at different x-values.
• In cases like this, it graphically demonstrates the approach towards \( \frac{1}{2} \) as \( x \to 0 \).
• It aids in discovering any asymptotic behavior or discontinuities.

Using graphing technology or software, like a graphing calculator, can further supplement your understanding by clearly showing the trend in function values, reinforcing the limit found analytically through methods like L'Hopital's Rule.