Problem 76 - 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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x^{4}-1}{x^{2}-1}$$

Step-by-Step Solution

Answer

Question: Find the horizontal and vertical asymptotes of the function \(f(x) = \frac{x^4 - 1}{x^2 - 1}\). Answer: The function \(f(x) = \frac{x^4 - 1}{x^2 - 1}\) has no horizontal or vertical asymptotes.

Step 1: Simplify and find vertical asymptotes

Factor: \(f(x) = \frac{x^4-1}{x^2-1} = \frac{(x^2+1)(x^2-1)}{x^2-1}\). For \(x \neq \pm 1\), \(f(x) = x^2+1\). The apparent discontinuities at \(x = \pm 1\) are removable (holes), so there are no vertical asymptotes.

Step 2: Find horizontal asymptotes

\(\lim_{x \to \pm\infty} f(x) = \lim_{x \to \pm\infty} (x^2+1) = \infty\). There are no horizontal asymptotes. The function behaves like \(y = x^2 + 1\) for large \(|x|\).

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input values (\(x\)) become very large or very small. They are particularly useful for understanding the behavior of functions as they extend towards positive or negative infinity. For rational functions, which are ratios of polynomials, we find horizontal asymptotes by looking at the degrees of the numerator and the denominator.

• 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 equals the degree of the denominator, the horizontal asymptote is at \(y =\) the ratio of leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there are no horizontal asymptotes, as seen in the function \(f(x) = \frac{x^{4} - 1}{x^{2} - 1}\). In this case, the graph heads towards infinity as \(x\) approaches \( \infty\) or \(-\infty\).

Vertical Asymptotes

Vertical asymptotes occur where a function grows indefinitely as it approaches a certain \(x\)-value. They often indicate values where a function is undefined or where division by zero might happen. For rational functions, check where the denominator equals zero since those are commonly the locations of vertical asymptotes.
However, these points need to remain undefined in the reduced form of the function, after canceling common factors of the numerator and denominator.
In our exercise, once we reduced the function \(f(x) = \frac{(x^2 + 1)(x - 1)(x + 1)}{(x - 1)(x + 1)}\) to \(x^2 + 1\), the removable discontinuities at \(x = 1\) and \(x = -1\) do not result in vertical asymptotes. This means there are no points where the function shoots to infinity, indicating no vertical asymptotes exist for this particular function.

Limits at Infinity

Limits at infinity help us understand how functions behave as \(x\) becomes extremely large or small. Evaluating \(\lim_{x \to \pm \infty} f(x)\) provides insights into the long-term behavior of the function. In the given exercise, computing limits as \(x\) approaches infinity or negative infinity shows that the function \(f(x)\) grows indefinitely.

• To compute these limits, first, simplify the function by dividing the terms by the highest power found in the denominator.
• Once simplified, evaluate the limits by considering the behavior of the leading terms in the numerator and the denominator.
• In this exercise, as \(x\) heads towards positive or negative infinity, the function tends to behave like \(x^2\), leading to infinite values.

Remember, if limits of a function at infinity yield a real number, a horizontal asymptote exists. Here, however, the limits being infinity confirm no horizontal asymptote.

Rational Functions

Rational functions are quotients involving polynomials, expressed as \(\frac{P(x)}{Q(x)}\). These functions are defined where the denominator \(Q(x)\) is not zero and can exhibit different behaviors based on the degrees of the polynomial expressions in the numerator \(P(x)\) and denominator \(Q(x)\).
Key points to note include:

• Asymptotic behavior often stems from the polynomial degree differences between the numerator and denominator.
• Vertical asymptotes generally occur where the rational function becomes undefined due to a zero in the denominator.
• Horizontal or oblique asymptotes can be determined based on the degree comparisons, as mentioned earlier.

In our case, simplification by canceling common factors was crucial in determining the actual function behavior without unnecessary complexities. The simplification showed that \(f(x)\), after canceling terms, leads to a polynomial expression without asymptotes.