Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

\(\lim_{x \rightarrow -3} \sqrt{2 x^{4}+x^{2}} = \sqrt{171}\)

Step 1: Identify the function

First, let's identify the function we are working with: \(f(x) = \sqrt{2x^4 + x^2}\) We are asked to find \(\lim_{x\rightarrow-3} f(x)\).

Step 2: Substitute

Now, we will substitute \(x=-3\) into the function: \(f(-3) = \sqrt{2(-3)^4 + (-3)^2}\)

Step 3: Evaluate the expression

Now, let's evaluate the expression: \(f(-3) = \sqrt{2(81) + 9}\) \(f(-3) = \sqrt{162 + 9}\) \(f(-3) = \sqrt{171}\)

Step 4: Write the limit

Now that we have the value of the function at \(x=-3\), we can write the limit: \(\lim_{x\rightarrow-3} \sqrt{2x^4+x^2} = \sqrt{171}\)

Key Concepts

Evaluating Limits

Evaluating limits is a fundamental concept in calculus that helps us understand the behavior of functions as their inputs approach a certain value. In this exercise, we want to find the limit of the function \( f(x) = \sqrt{2x^4 + x^2} \) as \( x \) approaches \(-3\).

• The first step in evaluating the limit is to understand the nature of the function involved. Here, the function includes a radical expression.
• Next, substitute the value \( x = -3 \) directly into the function. This is often the simplest way to evaluate limits, especially when the function is continuous around the value you are approaching.
• Finally, perform the algebraic simplifications inside the square root and solve it to get the numerical value.

Evaluating limits by direct substitution works smoothly for continuous functions, but sometimes a more sophisticated approach like factoring or rationalizing is required if substituting directly leads to an indeterminate form.

Calculus Functions

Calculus functions, such as polynomials, exponentials, and radicals, describe how quantities change and are crucial for modeling real-world scenarios. In this exercise, the function \( f(x) = \sqrt{2x^4 + x^2} \) is a polynomial under a radical, which is continuous and differentiable at all points in its domain.

• Functions like \( f(x) \) can be broken down into simpler parts — here it's helpful to focus on whether each component inside the square root is well-behaved.
• Given its form, for any real number \( x \), calculating \( f(x) \) involves finding even powers, which are inherently non-negative, allowing the square root to produce a real number.
• While handling radical functions, ensuring that expressions under the square root stay non-negative across your area of interest is crucial to avoid any undefined terms.

Understanding the characteristics of calculus functions facilitates easier limit evaluation. In this context, because the function is a combination of polynomial terms, substitutions and evaluations remain straightforward.

Limit of a Function

The limit of a function refers to the value that a function approaches as the input approaches a certain point. It is represented as \( \lim_{x \to a} f(x) \), where \( a \) is the point of interest.

• For \( f(x) = \sqrt{2x^4 + x^2} \), the task was to find \( \lim_{x \to -3} f(x) \). Here, the approach involved substituting \( x = -3 \) directly into \( f(x) \).
• The result \( \sqrt{171} \) indicates that as \( x \) nears \(-3\), the function value closely approaches \( \sqrt{171} \).
• Limits provide a formal way of discussing the behavior of functions at points that might be difficult to calculate directly, such as discontinuities or infinities.

By evaluating a limit, we gain insight into predictable behavior of functions in contexts where direct evaluation is not straightforward, though direct substitution proved effective in this situation to establish the function's approaching value.