Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}}\)

Step-by-Step Solution

Answer

The short answer is: \(\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}} = 2\).

Step 1: Determine if direct substitution works

: First, try to substitute the value of x directly into the given function. We will use the following function: \(f(x) = \sqrt{\frac{2x^3 + 4}{x^2 + 1}}\) By substituting x = 2, we get: \(f(2) = \sqrt{\frac{2(2^3) + 4}{(2^2) + 1}} = \sqrt{\frac{20}{5}}\) Since the function is defined at x = 2, we can directly substitute the value.

Step 2: Calculate the limit

: Since we found out that direct substitution works, we can now find the limit of the given function as x approaches 2 using the limit laws. The given function is: \(\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}}\) By substituting x = 2, we get: \(\lim _{x \rightarrow 2} \sqrt{\frac{2(2^3) + 4}{(2^2) + 1}}\) After calculating the expression inside the square root, we get: \(\lim _{x \rightarrow 2} \sqrt{\frac{20}{5}}\) Now, simplify the expression under the square root: \(\lim _{x \rightarrow 2} \sqrt{4}\) Finally, find the square root: \(\lim _{x \rightarrow 2} 2\)

Step 3: State the conclusion

: Since we were able to calculate the limit of the function directly, the limit of the given function as x approaches 2 is: \(\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}} = 2\)

Key Concepts

Limits

In calculus, limits help us understand the behavior of a function as it approaches a certain point. A limit answers the question: "What value is the function getting close to as the input value (usually denoted as \(x\)) approaches a certain point?" This concept is foundational in calculus and is crucial in defining derivatives and integrals. When considering limits, we do not always need to concern ourselves with what is happening at the point itself. Instead, we are interested in the tendencies or trends as we get arbitrarily close to the given point. For instance, in our exercise, we were interested in the behavior of the function \( f(x) = \sqrt{\frac{2x^3 + 4}{x^2 + 1}} \) as \(x\) approaches 2. Understanding limits is essential because they provide a way to work with functions that are not well-defined at certain points, such as in cases of singularities or undefined expressions. Moreover, limits can help predict the behavior of phenomena modeled by mathematical functions.

Direct Substitution

Direct substitution is often the most straightforward method to find the limit of a function as \(x\) approaches a specific value. This technique involves replacing the variable \(x\) with the value that it approaches in the expression. If the resulting expression is defined and finite, the limit exists and is equal to the value calculated using direct substitution. For example, in our exercise, we substituted \(x = 2\) directly into the function \( f(x) = \sqrt{\frac{2x^3 + 4}{x^2 + 1}} \). After substitution, we found that the expression was defined:

• The numerator \(2(2^3) + 4\) simplifies to 20.
• The denominator \((2^2) + 1\) simplifies to 5.

By the calculation \( \sqrt{\frac{20}{5}} = \sqrt{4} = 2 \), we see the limit is achieved through direct substitution. Direct substitution can simplify the process significantly when it applies, avoiding more complex algebraic manipulation. It is always the first technique to attempt when dealing with limits.

Limit Laws

Limit laws are rules that help simplify finding limits, especially when direct substitution is not feasible. These laws provide techniques to split, combine, and manipulate the different parts of an expression to make it easier to evaluate. Here are some essential limit laws:

• Sum/Difference Law: The limit of a sum/difference is the sum/difference of the limits.
• Product Law: The limit of a product is the product of the limits.
• Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.
• Power Law: The limit of a power is the power of the limit.

These laws help us break down complex expressions into manageable pieces. In our specific example, once the direct substitution was confirmed to work, applying limit laws verified that the step-by-step manipulation was consistent, ensuring accuracy in the final calculation of the limit.