Question

Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptote(s) of \(f\) (if any). $$f(x)=\frac{4 x^{3}+1}{2 x^{3}+\sqrt{16 x^{6}+1}}$$

Short answer

Answer: The horizontal asymptote of the function is \(y = 2\).

Step-by-step solution

Evaluating the limit as x approaches infinity

To find the limit as x approaches infinity, analyze the behavior of the rational function. One way of finding this limit is to divide the numerator and denominator by the highest power of x present in the function, which is \(x^3\) in this case. $$\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{4 x^{3}+1}{2 x^{3}+\sqrt{16 x^{6}+1}}$$ $$\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{4+\frac{1}{x^{3}}}{2+\frac{\sqrt{16+ \frac{1}{x^{3}}}}{x^{3}}}$$ As x approaches infinity, \(1/x^3\) approaches 0. Therefore, $$ \lim _{x \rightarrow \infty} f(x) = \frac{4}{2}\implies \lim _{x \rightarrow \infty} f(x) = 2$$

Evaluating the limit as x approaches negative infinity

To find the limit as x approaches negative infinity, we can use the same method as before, dividing the numerator and denominator by the highest power of x present in the function. $$\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{4 x^{3}+1}{2 x^{3}+\sqrt{16 x^{6}+1}}$$ $$\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{4+\frac{1}{x^{3}}}{2+\frac{\sqrt{16+ \frac{1}{x^{3}}}}{x^{3}}}$$ As x approaches negative infinity, \(1/x^3\) approaches 0. Therefore, the limit is: $$ \lim _{x \rightarrow -\infty} f(x) = \frac{4}{2}\implies \lim _{x \rightarrow -\infty} f(x) = 2$$

Finding the horizontal asymptote(s)

A horizontal asymptote is a horizontal line that the function approaches as x approaches infinity or negative infinity. In this case, since the limits as x approaches infinity and negative infinity are the same value, 2, there is a single horizontal asymptote: \(y = 2\).

Key concepts

Limits at Infinity

When we talk about limits at infinity, we're examining how a function behaves as the input (often denoted as \(x\)) becomes very large or very small either positively or negatively.
It's like asking, "What happens to the value of the function as \(x\) gets extremely big or moves far to the negative side?"
These limits help us understand the long-term behavior of functions, particularly rational functions.

• As \(x \to \, +\infty\): We check how the function behaves as \(x\) grows without end in the positive direction.
• As \(x \to \, -\infty\): Similarly, we evaluate how the function responds when \(x\) stretches infinitely in the negative direction.

Understanding these concepts allows us to figure out potential horizontal asymptotes, which show the line that the function seems to "settle" into as it reaches extreme values. In our solution, the limit both as \(x\) approaches \(+\infty\) and \(-\infty\) was 2, suggesting a horizontal asymptote at \(y = 2\).
Knowing how to calculate these limits provides insight into the stable behavior of functions, especially in real-world applications where extreme conditions are analyzed.

Rational Functions

Rational functions are a special category of functions that are the ratio of two polynomials. This means they take the form:
\[ \frac{P(x)}{Q(x)} \]
where both \(P(x)\) and \(Q(x)\) are polynomials.
These functions showcase a lot of interesting properties, largely due to the existence of numerators and denominators.

• Polynomials: Each polynomial in the numerator and denominator has its own degree (the highest power of \(x\) present). The degrees of these polynomials help us understand the behavior of the rational function, particularly as \(x\) moves towards infinity or negative infinity.
• Vertical Asymptotes: Occur when the denominator equals zero at some point (assuming the numerator is not zero at that point as well), causing the function to blow up (become undefined or approach infinity).
• Horizontal Asymptotes: These depend on the leading coefficients and the degrees of the polynomials in the numerator and denominator, as we've explored in the limits at infinity.

Rational functions often require careful manipulation when evaluating their limits, such as dividing every term by the highest power of \(x\) found in the denominator. By simplifying rational expressions, as done in our exercise, we can discover the long-term behavior of the function, which is crucial for understanding its graph and asymptotic properties.

Evaluating Limits

Evaluating limits is a fundamental skill in calculus, where we determine the value that a function "approaches" as the input approaches some point. The process can become pivotal in understanding many forms of behavior in a function, such as continuity and asymptotes.
The key steps for evaluating limits at infinity, especially for rational functions, are:

• Identify the Degrees: Figure out the highest power of \(x\) present in the function, both in the numerator and the denominator.
• Divide by the Highest Power: Adjust the expression by dividing each term by this highest power of \(x\). This often simplifies the function into a more manageable form.
• Substitute Limits: Carefully substitute limits into the simplified form, noting that terms like \(1/x^n\) will approach zero as \(x\) extends towards infinity or negative infinity.

In our given problem, the highest power was \(x^3\). By dividing both the numerator and the denominator by \(x^3\), we could easily forego complex expressions and simplify our solutions to reach the constant that defines the horizontal asymptote, \(y = 2\).
Mastering these techniques not only helps tackle complex problems but also builds a deeper understanding of how functions behave over large ranges.