Problem 3 - Calculus of a Single Variable

Question

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} $$

Step-by-Step Solution

Answer

The limit as x approaches 0 is \( \frac{1}{2}\).

Step 1: Rationalize the numerator

The first step is to rationalize the numerator of the expression that is \((\sqrt{x+1}-1)\). This is done by multiplying the numerator and the denominator by the conjugate of the numerator \( (\sqrt{x+1}+1)\). This would give: \[ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} * \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} \]

Step 2: Simplify the expression

The next step is to simplify the expression. When we multiply out the numerator, we get \( (x+1) - 1 = x \), and the denominator becomes \( x(\sqrt{x+1}+1) \). This simplification will result in: \[ \lim _{x \rightarrow 0} \frac{x}{x(\sqrt{x+1}+1)} \]

Step 3: Cancel out the common terms

There's a common term \(x\) in the numerator and denominator which can be cancelled out. This would give: \[ \lim _{x \rightarrow 0} \frac{1}{\sqrt{x+1}+1} \] This is a simpler expression which is easy to solve directly by substituting \(x=0\)

Step 4: Evaluate the Limit

After simplifying, substitute the value of \(x = 0 \) into the simplified expression from Step 3: \[ \lim _{x \rightarrow 0} \frac{1}{\sqrt{x+1}+1} = \frac{1}{\sqrt{0+1}+1} = \frac{1}{2} \]

Key Concepts

Estimating Limits Numerically

When tackling limits, starting with a numerical estimation can be very insightful. Numerical estimation involves selecting values that approach the limit of interest, in this case, as \(x\) approaches 0. By inputting values that get successively closer to 0 from both the negative and positive sides, you can observe how the function behaves. This process creates a table of values which reveals the trend and helps in predicting what the outcome should be as \(x\) approaches the limit point.

• Choose several \(x\) values near 0, such as \(x = -0.1, -0.01, 0.01, 0.1\).
• Calculate the corresponding function values \(\frac{\sqrt{x+1}-1}{x}\) for these \(x\) values.
• Observe the result and estimate where it tends as \(x\) approaches 0.

This method provides a foundational understanding before attempting algebraic simplification or confirmation using a graphing tool.

Rationalizing Expressions

Rationalizing involves eliminating any irrational components, such as square roots, from the numerator or denominator. In this example, we start by focusing on the numerator \(\sqrt{x+1} - 1\). By multiplying both the numerator and the denominator by the conjugate \(\sqrt{x+1} + 1\), we effectively turn the difference of squares into a rational expression.
This step is critical because it simplifies the process and allows for easier manipulation and calculation:

• Multiply: \(\left(\sqrt{x+1} - 1\right) \cdot \left(\sqrt{x+1} + 1\right) = (x+1) - 1\).
• This neatens to a straightforward \(x\) in the numerator.
• Thus, \(\frac{\sqrt{x+1}-1}{x} \cdot \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} = \frac{x}{x(\sqrt{x+1}+1)}\).

Simplifying to rational forms is pivotal in making calculations more straightforward, enabling further algebraic techniques.

Simplifying Expressions

Simplifying expressions helps make the limit evaluation achievable. Here, after rationalizing the numerator, the expression becomes \(\frac{x}{x(\sqrt{x+1}+1)}\). By inspecting this, it's evident that the \(x\) in the numerator and denominator can cancel each other out, assuming \(x eq 0\), leaving us with \(\frac{1}{\sqrt{x+1}+1}\).

• Cancelling terms is permissible when they are common in both the numerator and denominator.
• This practice reduces the expression to its simplest form, providing easier insight into evaluating the limit.

Upon simplification, substituting can then be directly performed, and complex calculations are avoided.

Graphing Utility

A graphing utility like Desmos or a scientific calculator can visually confirm the limit evaluation. By graphing \(y = \frac{\sqrt{x+1}-1}{x}\), you can observe the behavior around \(x = 0\). This step offers visual confirmation and reinforces numerical and analytical findings.

• Enter the expression into the graphing tool.
• Zoom into the areas around \(x = 0\) to observe any trends or paths the graph follows.
• Check if the graph appears to approach a value of \(\frac{1}{2}\) as \(x\) nears 0.

Using a graphing utility provides a graphical perspective that highlights the limit behavior, confirming the computational results.

Substitution Method in Limits

Sometimes, especially after simplification, substituting the value of \(x\) directly is the most straightforward way to determine the limit. In our case, once the expression is simplified to \(\frac{1}{\sqrt{x+1}+1}\), you can easily substitute \(x = 0\) to find the limit.

• Insert \(x = 0\) into \(\frac{1}{\sqrt{x+1}+1}\).
• This results in \(\frac{1}{\sqrt{0+1}+1} = \frac{1}{2}\).
• The limit is therefore \(\frac{1}{2}\).

The substitution method is effective when expressions are rationalized or simplified enough to avoid indeterminate forms, solidifying numerical and graph-based conclusions.