Problem 49 - Calculus: Early Transcendentals

Question

Horizontal asymptotes Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptotes of \(f(\text {if any})\). $$f(x)=\frac{\sqrt[3]{x^{6}+8}}{4 x^{2}+\sqrt{3 x^{4}+1}}$$

Step-by-Step Solution

Answer

$$f(x)=\frac{\sqrt[3]{x^{6}+8}}{4 x^{2}+\sqrt{3 x^{4}+1}}$$ Answer: The horizontal asymptote for the given function is $$y=\frac{1}{\sqrt{3}}$$.

Step 1: Analyze the function

We have the function: $$f(x)=\frac{\sqrt[3]{x^{6}+8}}{4 x^{2}+\sqrt{3 x^{4}+1}}$$ As x approaches positive or negative infinity, the highest powers of x in both numerator and denominator will become dominant terms.

Step 2: Find dominant terms

In the numerator, the dominant term is \(x^6\). In the denominator, the dominant term is \(x^4\). So we can rewrite the function as: $$f(x) \approx \frac{x^2\sqrt[3]{x^{6}}}{x^{4}\sqrt{3 x^{4}}}$$ Then we can simplify the above function: $$f(x) \approx \frac{x^2}{x^2\sqrt{3}} = \frac{1}{\sqrt{3}}$$

Step 3: horizontal asymptote as x → +∞ and x → -∞

Now we can find the limits of the simplified function as x approaches positive and negative infinity: $$\lim_{x\rightarrow \infty} \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}}$$ $$\lim_{x\rightarrow -\infty} \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}}$$ Since the limits exist and are the same, there is a horizontal asymptote: $$y=\frac{1}{\sqrt{3}}$$

Key Concepts

Limits at Infinity

Understanding limits at infinity is key to identifying horizontal asymptotes in functions. When we talk about limits at infinity, we are asking what value a function approaches as the variable, usually 'x', goes off towards negative or positive infinity. Think of it as following the function's graph farther and farther along the horizontal axis and observing where it 'settles down' as it gets very far away.

In our exercise, we are dealing with finding \(\lim_{x\rightarrow \infty} f(x)\) and \(\lim_{x\rightarrow -\infty} f(x)\). To simplify, we only consider the most dominant terms in the numerator and denominator, as those vastly outpace any other terms when x grows very large in magnitude. Once simplified, we can easily see the behavior at infinity, providing us with the function's horizontal asymptote.

Asymptotic Behavior

The asymptotic behavior of a function describes how the function behaves as it moves towards infinity or negative infinity. Specifically, a horizontal asymptote refers to a horizontal line that the function approaches but does not cross as x increases or decreases without bound. It’s like a boundary that the function gets infinitely close to but never actually touches.

In simpler terms, if the function f(x) gets closer and closer to a specific y-value as x becomes very large or very small, that y-value is the horizontal asymptote of the function. Our function's asymptotic behavior, as x goes to infinity and negative infinity, tells us that the horizontal asymptote of the function is \(y=\frac{1}{\sqrt{3}}\).

Dominant Terms in Functions

When analyzing functions for their end behavior, it’s the dominant terms that matter. These are the terms in a function that grow the fastest when the variable increases or decreases in magnitude. They quite literally dominate the function's behavior as we head towards infinity or negative infinity, dwarfing the influence of all other terms.

For our cubic root function, \(\frac{\sqrt[3]{x^{6}+8}}{4x^{2}+\sqrt{3x^{4}+1}}\), the dominant terms are \(x^6\) in the numerator and \(x^4\) in the denominator. By focusing on these dominant terms, we can simplify the expression to see the general trend as x grows very large or becomes a very large negative number, making it easier to pinpoint the horizontal asymptote.

Simplifying Expressions

The process of simplifying expressions involves reducing a complex mathematical expression into its simplest form. This can mean combining like terms, reducing fractions, or, in the context of limits at infinity, collapsing a function down to its most significant terms. This is often the most crucial step in identifying horizontal asymptotes since it allows us to cut through the complexity and see what value the function is tending toward.

In the given function, by simplifying \(\frac{x^2\sqrt[3]{x^{6}}}{x^{4}\sqrt{3x^{4}}}\) to \(\frac{1}{\sqrt{3}}\), we drastically reduced the complexity of the expression, making it clear that the limit—and thus the horizontal asymptote—is \(\frac{1}{\sqrt{3}}\). Simplification is a powerful tool in calculus and especially in analyzing the behavior of functions at their extremes.