Problem 35 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-3^{+}}\left(\frac{|x+3|}{x+3}+\sqrt{x+3}+1\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the given function as x approaches -3 from the positive side is 2.

Step 1: Identify the three terms in the function

We will analyze each term in the given function separately. The three terms are: 1. \(\frac{|x+3|}{x+3}\) 2. \(\sqrt{x+3}\) 3. \(1\) We will start by analyzing the behavior of the first term as x approaches -3 from the positive side.

Step 2: Analyze the behavior of the first term as x approaches -3 from the positive side

When x is close to -3+, the expression inside the absolute value |x+3| is very close to 0, but it is positive. Therefore, we can rewrite the first term as: $$\frac{x+3}{x+3}$$ Note that as x approaches -3 from the positive side, this term simplifies to: $$\lim _{x \rightarrow -3^{+}}\left(\frac{x+3}{x+3}\right) = 1$$ Now, let's analyze the second term.

Step 3: Analyze the behavior of the second term as x approaches -3 from the positive side

As x approaches -3 from the positive side, the expression inside the square root, x+3, approaches 0. The second term becomes: $$\lim _{x \rightarrow -3^{+}}\sqrt{x+3} = \sqrt{0} = 0$$ Now, we will analyze the third term.

Step 4: Analyze the behavior of the third term as x approaches -3 from the positive side

Since the third term is a constant, its limit will be the same constant: $$\lim _{x \rightarrow -3^{+}} 1 = 1$$

Step 5: Combine the three terms to find the limit

Now that we've found the limits of each term as x approaches -3 from the positive side, we can combine them to find the limit of the entire function: $$\lim _{x \rightarrow -3^{+}}\left(\frac{|x+3|}{x+3}+\sqrt{x+3}+1\right) = 1 + 0 + 1$$ Therefore, the limit of the given function as x approaches -3 from the positive side is: $$\lim _{x \rightarrow -3^{+}}\left(\frac{|x+3|}{x+3}+\sqrt{x+3}+1\right) = \boxed{2}$$

Key Concepts

One-Sided Limit

A one-sided limit focuses on what happens to a function as the variable approaches a certain value from one side only. In this exercise, we are considering the limit as \( x \) approaches \(-3\) from the positive side, written as \( x \to -3^+ \). This means we are examining values of \( x \) that are slightly more than \(-3\), like \(-2.9\) or \(-2.99\), and observing how the function behaves at these points.

Understanding one-sided limits is crucial when functions behave differently from the left and the right of a certain point. By addressing from one side, you can see specific trends or behaviors that are not apparent when looking from both directions simultaneously.

• Define the direction: \( ^+ \) or \( ^- \) helps to narrow focus.
• Useful for investigating discontinuities or sharp corners in graphs.
• Important for precise mathematical analysis in calculus.

Absolute Value Function

The absolute value function \( |x| \) provides the distance of \( x \) from zero, always resulting in a non-negative number. In this problem, you see it as \( |x+3| \), shifting the focus to how far \( x \) is from \(-3\).

When \( x \) is close to but greater than \(-3\), \( x+3 \) is a small positive number. As a result, \( |x+3| = x+3 \), simplifying the expression. This makes the fraction \( \frac{|x+3|}{x+3} \) turn into 1.

Understanding absolute values helps in dealing with expressions that can change sign:

• It ignores sign changes by focusing on magnitude.
• Simplifies expressions for analysis in limits and equations.
• Essential for understanding distance-related problems.

Square Root Function

The square root function involves taking the principal (non-negative) root of a number, denoted as \( \sqrt{x} \). In our function, it's part of the expression \( \sqrt{x+3} \).

As \( x \) approaches \(-3\) from the right, \( x+3 \) is positive and gets closer to zero. Therefore, \( \sqrt{x+3} \) approaches \( \sqrt{0} = 0 \).

Key points about the square root function:

• Non-negative input yields non-negative output.
• Near zero inputs result in outputs close to zero.
• Important for working with certain limit and continuity problems.

This function helps us describe settings where small input changes affect output, particularly as we approach boundary values.

Function Limits

Function limits help us understand the behavior of functions as they approach a certain point from different directions. For a function \( f(x) \), the limit as \( x \to a \) is what \( f(x) \) gets closer to as \( x \) nears \( a \).

In our exercise, the limit of the entire function as \( x \) approaches \(-3^+\) is the sum of the individual limits of its parts. We determined:

• \( \lim_{x \to -3^+} \frac{|x+3|}{x+3} = 1 \)
• \( \lim_{x \to -3^+} \sqrt{x+3} = 0 \)
• \( \lim_{x \to -3^+} 1 = 1 \)

By adding these together, we get \( 1 + 0 + 1 = 2 \).

Limits encapsulate important information about function behavior near specific points, even if the function isn't defined precisely there: