Problem 11 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow \sqrt{5}}\left(9-x^{2}\right)^{-5 / 2} $$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{32}\).

Step 1: Identify the Form of the Function

The given function is \((9 - x^2)^{-5/2}\). We need to identify how the expression behaves as \(x\) approaches \(\sqrt{5}\).

Step 2: Substitute the Limit Value

Substitute \(x = \sqrt{5}\) into the expression \(9 - x^2\):\[9 - (\sqrt{5})^2 = 9 - 5 = 4\]

Step 3: Compute the Expression

After substitution, the expression becomes \((4)^{-5/2}\). We need to evaluate this expression.

Step 4: Simplify the Power

Calculate \(4^{-5/2}\) using the properties of exponents. This equals:\[(4^{-5/2}) = (2^2)^{-5/2} = 2^{-5} = \frac{1}{2^5} = \frac{1}{32}\]

Step 5: Conclusion of the Limit

Since we successfully evaluated the expression, the limit \(\lim_{x \to \sqrt{5}} (9-x^2)^{-5/2}\) evaluates to \(\frac{1}{32}\).

Key Concepts

Limit Evaluation

Limit evaluation refers to calculating the value that a function approaches as the input approaches some value. In this scenario, you are dealing with the expression \( \lim_{x \rightarrow \sqrt{5}}(9-x^{2})^{-5 / 2} \). To evaluate this limit, you follow a method of substitution where you replace \( x \) with \( \sqrt{5} \) in the expression.
By doing so, you calculate the value that the function approaches, thus obtaining the correct limit. This process is pivotal since understanding limits forms a cornerstone of calculus. Limits help us to understand how functions behave, especially near points that are not necessarily part of the function's domain.

Properties of Exponents

The properties of exponents are mathematical rules that govern how expressions involving powers can be simplified or manipulated. These properties are crucial in evaluating the function \( (4)^{-5/2} \).
Key exponent rules include:

• Power of a Power: \((a^b)^c = a^{b \cdot c} \).
• Negative Exponent: \( a^{-b} = \frac{1}{a^b} \).
• Fractional Exponents: \( a^{m/n} = \sqrt[n]{a^m} \).

Using these rules in the example, you express \( (4)^{-5/2} \) as \( (2^2)^{-5/2} \). This simplifies to \( 2^{-5} \), which further evaluates to \( \frac{1}{2^5} \). These properties enable you to transform complex expressions into simpler, solvable forms. Understanding them can make dealing with exponents much less daunting.

Substitution Method

The substitution method in calculus involves replacing a variable with a particular value to simplify an evaluation process. Here, you substitute \( x = \sqrt{5} \) to evaluate the limit.
This substitution transforms the expression from \( (9-x^{2}) \) to \( 9 - (\sqrt{5})^2 \), simplifying it to \( 9 - 5 = 4 \).
The main advantage of substitution is that it converts a problem that might seem complex into a basic computation of known quantities. This technique is widely used because of its simplicity and effectiveness in determining values of limits when direct computation might otherwise be difficult.

Simplification in Calculus

Simplification in calculus often involves reducing expressions to their simplest form to either find limits, derivatives, or integrals more straightforwardly. In this particular exercise, after substituting \( x = \sqrt{5} \), the expression reduces to \((4)^{-5/2} \).
Simplification here involves using properties of exponents to further break down the expression to \( \frac{1}{32} \).

• Convert \( (4)^{-5/2} \) using basic exponent rules.
• Understand how breaking complex forms into basic calculations helps solve the problem.

By simplifying expressions, you make them easier to handle and understand. This process is essential in calculus as it allows for the evaluation of mathematical problems that might be challenging in their original forms.