Problem 73 - Calculus: Early Transcendentals

Question

Horizontal and vertical asymptotes. a. Analyze \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\) analyze \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{3 x^{4}+3 x^{3}-36 x^{2}}{x^{4}-25 x^{2}+144}$$

Step-by-Step Solution

Answer

Answer: The horizontal asymptote is at \(y=3\), and the vertical asymptotes are at \(x=-3, -4, 3,\) and \(4\).

Step 1: Determine horizontal asymptotes

To find horizontal asymptotes, we will analyze the limits as \(x\) approaches \(\infty\) and \(-\infty\). Calculate \(\lim_{x\to\infty}f(x)\) and \(\lim_{x\to-\infty}f(x)\): Since both the numerator and denominator are polynomial functions with the same degree, we can find the horizontal asymptote by comparing the leading coefficients in both the numerator and the denominator: $$\lim_{x\to\infty}f(x) \ = \lim_{x\to-\infty}f(x) = \frac{3}{1} = 3$$ This indicates that there is a horizontal asymptote at \(y=3\).

Step 2: Determine vertical asymptotes

To find vertical asymptotes, we need to identify the values of \(x\) for which the denominator of \(f(x)\) is equal to zero: $$x^4 - 25x^2 + 144 = 0$$ Factor the quadratic polynomial with respect to \(x^2\): $$(x^2 - 9)(x^2 - 16) = 0$$ Now, find the solutions for \(x\): $$x^2-9=0 \Rightarrow x = \pm 3$$ $$x^2-16=0 \Rightarrow x = \pm 4$$ So, we have 4 possible vertical asymptotes at \(x = \{-3, -4, 3, 4\}\).

Step 3: Analyze the behavior near the vertical asymptotes

Now, we need to find the limits of \(f(x)\) as \(x\) approaches each of these vertical asymptotes from the left and the right: For \(x=-3\): $$\lim_{x \rightarrow -3^-} f(x) = -\infty$$ $$\lim_{x \rightarrow -3^+} f(x) = \infty$$ For \(x=-4\): $$\lim_{x \rightarrow -4^-} f(x) = \infty$$ $$\lim_{x \rightarrow -4^+} f(x) = -\infty$$ For \(x=3\): $$\lim_{x \rightarrow 3^-} f(x) = \infty$$ $$\lim_{x \rightarrow 3^+} f(x) = -\infty$$ For \(x=4\): $$\lim_{x \rightarrow 4^-} f(x) = -\infty$$ $$\lim_{x \rightarrow 4^+} f(x) = \infty$$ In conclusion, the function \(f(x)\) has a horizontal asymptote at \(y=3\) and vertical asymptotes at \(x=-3, -4, 3,\) and \(4\).

Key Concepts

Horizontal Asymptote

Horizontal asymptotes are important for understanding the end behavior of a function. They tell us what value a function approaches as the input (usually noted as \(x\)) becomes very large or very negative.
When dealing with rational functions, which are functions represented as the ratio of two polynomials, finding horizontal asymptotes involves comparing the degrees of the numerator and the denominator. Here is how it generally works:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \(y = 0\).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is found by taking the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, no horizontal asymptote exists.

In the given function \(f(x) = \frac{3x^4 + 3x^3 - 36x^2}{x^4 - 25x^2 + 144}\), since the degrees of the numerator and denominator are the same (both are 4), we look at the leading coefficients. This gives us the horizontal asymptote: \(y = 3\).

Vertical Asymptote

Vertical asymptotes occur at values of \(x\) where a function tends to infinity, either positive or negative. For rational functions, these points are determined by finding the values of \(x\) that make the denominator zero while the numerator is not zero.
In our given function, we set the denominator equal to zero and solve for \(x\): \(x^4 - 25x^2 + 144 = 0\).
This can be factored, giving us: \((x^2 - 9)(x^2 - 16) = 0\).
Solving for \(x\), we get \(x = \pm 3\) or \(x = \pm 4\).
This indicates potential vertical asymptotes at \(x = -4, -3, 3,\) and \(4\). However, it's crucial to analyze the function's behavior near these points to confirm they are indeed points of vertical asymptotes. For each of these points, the limits show behavior tending to infinity, further confirming that these are indeed vertical asymptotes.

Limits

Limits are a fundamental concept in calculus that help us understand the behavior of functions as they approach certain points or infinity. When we say \(\lim_{x \to a} f(x) = L\), this means that as \(x\) gets closer to \(a\), \(f(x)\) gets closer to \(L\).
We use limits to find both horizontal and vertical asymptotes.

• For horizontal asymptotes, limits at infinity tell us the end behavior of the function, such as \(\lim_{x \to \infty} f(x)\).
• For vertical asymptotes, we observe limits as \(x\) approaches specific values from either side, \(\lim_{x \to a^+} f(x)\) and \(\lim_{x \to a^-} f(x)\).

The limits we computed for the vertical asymptotes were evaluated from both the left and right, confirming the behavior towards infinity or negative infinity, indicating a vertical asymptote. Therefore, limits are crucial for thoroughly analyzing the behavior of functions near certain points or at infinity.

Polynomial Functions

Polynomial functions are mathematical expressions built from variables (like \(x\)) raised to whole-number exponents and combined using addition, subtraction, and multiplication. They are fundamental in algebra and calculus.
The general form of a polynomial function is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(x\) is a variable.
In the context of asymptotes, polynomial functions often appear in rational expressions (ratios of two polynomials).
The degrees of these polynomials (the highest powers of \(x\)) and their leading coefficients (coefficients of the highest powers) play a crucial role in determining horizontal asymptotes. This was observed when we compared the degrees and leading coefficients of our rational function's numerator and denominator.
Understanding these properties is essential, as it helps in easily identifying asymptotic behavior and other characteristics of more complex rational functions.