Problem 37 - Calculus with Concepts in Calculus

Question

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=\frac{x^{2}-20 x+50}{\sqrt{2 x^{4}-512}} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = 4, x = -4\); Horizontal asymptote: \(y = \frac{1}{\sqrt{2}}\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero, and the numerator is non-zero. For the function \(f(x) = \frac{x^{2}-20 x+50}{\sqrt{2 x^{4}-512}}\), we need to find when \(\sqrt{2x^4 - 512} = 0\). This requires solving \(2x^4 - 512 = 0\). Divide the entire equation by 2 to get:\[x^4 - 256 = 0\].\(x^4 = 256\) implies \(x = \pm 4\), considering positive and negative roots. Therefore, potential vertical asymptotes may occur at \(x = 4\) and \(x = -4\). However, to confirm, \(f(x)\) must not have simplified common factors with the numerator that cancel out these asymptotes.

Step 2: Simplify Function and Confirm Verticals

Next, check if the numerator \(x^{2} - 20x + 50 \) can be simplified at \(x = 4\) or \(x = -4\). Solving \(x^{2} - 20x + 50 = 0\) using the quadratic formula gives roots not equal to 4 or -4. Hence, the factors do not cancel out with the numerator. Therefore, vertical asymptotes exist at \(x = 4\) and \(x = -4\).

Step 3: Identify Horizontal Asymptotes

Horizontal asymptotes depend on the behavior of \(f(x)\) as \(x\to \infty\). Analyze the degrees of the numerator and denominator. The degree of the numerator \(x^2 - 20x + 50\) is 2, and under the square root in the denominator, the expression \(2x^4\), simplifies to a degree of 2 when split as \(x^2\sqrt{2x^2 - 256}\). Therefore, the degrees appear equal when \(x\to \infty\). Compare the leading coefficients, where the numerator has \(1\) and the denominator appears with \(\sqrt{2}\). Thus, the horizontal asymptote is \(y = \frac{1}{\sqrt{2}}\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes provide important insights about the behavior of a function at specific points. In simpler terms, vertical asymptotes are invisible lines that highlight where a function goes off to infinity—or very, very big values in either the positive or negative direction. Let's break this down using our function \[f(x) = \frac{x^{2}-20x+50}{\sqrt{2x^4 - 512}}\].

• Vertical asymptotes occur when the function has a denominator equal to zero, because dividing by zero isn’t allowed in math.
• However, the numerator must not also be zero at this same point or they could cancel each other out, eliminating the asymptote.

For the given function, we set \[\sqrt{2x^4 - 512} = 0\] leading to solving \[2x^4 - 512 = 0\].Through simplification, we find \[x^4 = 256\], which results in the solutions \[x = \pm 4\].After checking the numerator \[x^{2} - 20x + 50\], we confirm it doesn't nullify at these points, meaning no simplification occurs to cancel the potential vertical asymptotes. Therefore, we can confidently state that vertical asymptotes exist at \[x = 4\] and \[x = -4\].

Horizontal Asymptotes

A horizontal asymptote offers clues about how a function behaves as it reaches very large values of x, either in the positive or negative direction; essentially, how the function levels off. For horizontal asymptotes in our function, it's important to compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator.Here, the numerator \(x^2 - 20x + 50\) is of degree 2. Regarding the denominator, understand that the expression \(\sqrt{2x^4 - 512}\) simplifies to degrees as \(x^2\sqrt{2x^2 - 256}\), giving it degree 2 at leading terms as well.

• If the degree of the numerator and the denominator are the same, the horizontal asymptote is found by dividing the leading coefficients.

The leading coefficient of the numerator is 1, and for the denominator, we treat the leading term equivalent to \(\sqrt{2}\). Thus, the horizontal asymptote of our function is\[y = \frac{1}{\sqrt{2}}.\]

Function Behavior as x Approaches Infinity

Understanding the behavior of a function as x approaches infinity helps predict how the function will act for extremely large values. This is useful in many areas, like engineering or economics, to understand long-term trends or outcomes.As x becomes really large (positively or negatively), we analyze the terms of the function that have the greatest influence. In function \[ f(x) = \frac{x^{2}-20x+50}{\sqrt{2x^4 - 512}} \], as seen in horizontal asymptotes, the numerator and the root of the denominator each trend toward the behavior dictated by their highest-degree terms.

• The term that grows the largest, \(x^2\) in the numerator and \(x^2\) within the root, dominate the function's expression.
• This domination demonstrates that the function's values stabilize and continue trending towards the horizontal asymptote.

In this case, they reveal that the function stabilizes towards \[y = \frac{1}{\sqrt{2}}\], which effectively describes how \(f(x)\) behaves as x goes to infinity.