Problem 36 - Calculus and its applications

Question

Use the Limit Properties to find the following limits. If a limit does not exist, state that fact. $$ \lim _{x \rightarrow 3^{+}} \sqrt{x^{2}-9} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Function Inside the Square Root

The expression inside the square root is \( x^2 - 9 \). We need to analyze how this function behaves as \( x \) approaches 3 from the right (i.e., \( x \to 3^+ \)).

Step 2: Substitute Close Values

When \( x \) is slightly greater than 3, say \( x = 3.1 \), the expression becomes \( (3.1)^2 - 9 = 0.61 \), which is positive. This indicates the function inside the square root is positive for \( x > 3 \).

Step 3: Evaluating the Limit as x Approaches 3 from the Right

Since the value inside the square root approaches 0 as \( x \) approaches 3 from the right, and the square root function is continuous for non-negative values, the limit is \( \sqrt{0} \).

Step 4: Conclusion

Therefore, the limit is \( 0 \) because the expression \( \sqrt{x^2 - 9} \) tends towards \( \sqrt{0} \) as \( x \to 3^+ \).

Key Concepts

Limit Properties

Limits help us understand the behavior of functions as they approach certain points or infinity. They are crucial when dealing with expressions that might not be straightforward at a given point.
One of the core ideas behind limits is using limit properties to break down complex problems into simpler parts. Let's go through some of these properties:

• **Sum/Difference Rule:** The limit of a sum/difference is the sum/difference of their limits, i.e., \( \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) \).
• **Product Rule:** The limit of a product is the product of their limits, i.e., \( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \).
• **Quotient Rule:** The limit of a quotient is the quotient of their limits, provided the limit of the denominator isn't zero, i.e., \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \).
• **Root Rule:** The limit of a root is the root of their limit, applicable if the limit exists and the root index n is a positive integer, i.e., \( \lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)} \).

Understanding and applying these basic properties simplify the process of determining limits, enabling you to tackle even trickier problems with ease.

Continuity

Continuity is an essential concept in calculus and becomes particularly relevant when evaluating limits. A function is continuous at a point if the following conditions are met:
1. The function is defined at that point, 2. The limit of the function as it approaches the point equals the function's value at the point.
Mathematically, a function \( f(x) \) is continuous at \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).
In simple terms, you can think of continuity as a curve that you can draw without lifting your pencil from the paper. The absence of breaks, jumps, or holes in a function graph indicates its continuity.

• If a function is continuous in its domain, finding limits, especially at a specific point, is often straightforward.
• If a function isn't continuous, there may be a need for other mathematical approaches to determine its limit.
• For the square root function examined in the problem, its continuity is guaranteed for any value inside the square root that is non-negative.

Square Root Function

The square root function is fundamental and commonly appears in calculus problems. It is defined as \( \sqrt{x} \), representing a number that, when squared, equals \( x \).
The square root function is continuous for all non-negative inputs (i.e., for \( x \geq 0 \)), which allows limits involving square roots to be easily assessed if their domain is respected.

• As in the given exercise problem, there is an interest in understanding how the function approaches a certain value, specifically when \( x \to 3^+ \).
• It is known that for the expression \( x^2 - 9 \), as \( x \) approaches 3 from the right, the expression inside \( \sqrt{x^2 - 9} \) becomes non-negative, allowing the evaluation of the square root.
• This continuous behavior specifies that as \( x \) nears 3, the entire function heads towards a calculable and predictable limit—0 in the context of this problem.

The understanding of how the square root function behaves, especially at critical points of interest, aids in solving calculus problems with greater confidence and accuracy.