Problem 45 - 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 \(f(\text {if any})\). $$f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}}$$

Step-by-Step Solution

Answer

Answer: No, there are no horizontal asymptotes for the function \(f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}}\) because the limits as x approaches positive infinity and negative infinity do not exist.

Step 1: Analyzing the function

Observe the function $$f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}}.$$ We notice that the denominator has an \(x^4\) and \(x^8\) term. The highest power of x is 8.

Step 2: Simplify the function

Factor out \(x^4\) from the denominator to simplify \(f(x)\): $$f(x)=\frac{1}{x^4 \left(2-\sqrt{4-\frac{9}{x^4}}\right)}.$$

Step 3: Find limit as x approaches infinity

Now, let's find the limit as \(x\) approaches infinity: $$\lim_{x \rightarrow \infty} f(x)= \lim_{x \rightarrow \infty} \frac{1}{x^4 \left(2-\sqrt{4-\frac{9}{x^4}}\right)}.$$ Since we're considering the limit as \(x\) approaches infinity, the term \(\frac{9}{x^4}\) will approach 0, because the denominator becomes increasingly larger. Therefore, we can rewrite the limit as: $$\lim_{x \rightarrow \infty} f(x)= \lim_{x \rightarrow \infty} \frac{1}{x^4 (2-\sqrt{4})}.$$ This limit simplifies to: $$\lim_{x \rightarrow \infty} f(x)= \lim_{x \rightarrow \infty} \frac{1}{x^4 (2-2)}.$$ The denominator has a 0 in it, so it simplifies to: $$\lim_{x \rightarrow \infty} f(x)= \lim_{x \rightarrow \infty} \frac{1}{0}.$$ This limit does not exist, so there is no horizontal asymptote as \(x \rightarrow \infty\).

Step 4: Find the limit as x approaches negative infinity

By using the same logic, we can find the limit as \(x\) approaches negative infinity. We see that the terms \(\frac{9}{x^4}\) will still approach 0 because when a negative number is raised to an even power it becomes positive. Therefore, $$\lim_{x \rightarrow -\infty} f(x)= \lim_{x \rightarrow -\infty} \frac{1}{x^4 \left(2-\sqrt{4-\frac{9}{x^4}}\right)}.$$ Following the same process, we get $$\lim_{x \rightarrow -\infty} f(x)= \lim_{x \rightarrow -\infty} \frac{1}{x^4 (2-\sqrt{4})}.$$ This simplifies to $$\lim_{x \rightarrow -\infty} f(x)= \lim_{x \rightarrow -\infty} \frac{1}{x^4 (2-2)}.$$ The denominator has a 0 in it, so it simplifies to:$$\lim_{x \rightarrow -\infty} f(x)= \lim_{x \rightarrow -\infty} \frac{1}{0}.$$ This limit does not exist, so there is no horizontal asymptote as \(x \rightarrow -\infty\).

Step 5: Conclusion

There are no horizontal asymptotes for the function $$f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}},$$ because the limits as x approaches positive infinity and negative infinity do not exist.

Key Concepts

Limits at Infinity

When studying the behavior of a function as the variable approaches extremely large positive or negative values, we delve into the concept of limits at infinity. In essence, determining the limit of a function f(x) as x approaches infinity—or negative infinity—is a way of describing the trend or end behavior of the function.

For the given function \( f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}} \), identifying the limits as x tends towards infinity in either direction involves simplifying the function where possible and observing the effects of increasingly large values of x on the function's value. The goal is to establish what value f(x) gets closer to as x becomes very large or very small in magnitude.

In our case, although the function simplifies to an indeterminate form of \( \frac{1}{0} \) when approaching both positive and negative infinity, this is indicative of the fact that the function does not level off at a particular value. Thus, no horizontal asymptote is present. This is crucial for students to understand, as not every function exhibits such asymptotic behavior.

Limit of a Function

Exploring the limit of a function is a fundamental concept in calculus, which allows us to understand the value that a function f(x) approaches as x gets close to a particular point. Limits do not always equal the function's value at that point, which is essential when dealing with undefined points or infinity.

With the function \( f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}} \), as attempted in our exercise, we investigate the limit by addressing the highest power of x in the denominator, which dictates the behavior of the function as x grows larger. As students, it's important to recognize that limits can result in real numbers, infinity, negative infinity, or they may not exist at all, depending on the structure and components of the function.

Asymptotic Behavior

The concept of asymptotic behavior describes how a function behaves as it moves closer to a line that it will never actually reach, known as an asymptote. A horizontal asymptote, in particular, is indicative of a function approaching a constant value as x goes to positive or negative infinity.

The exercise provided reveals that some functions, like ours \( f(x)=\frac{1}{2 x^{4}-\sqrt{4 x^{8}-9 x^{4}}} \), do not have horizontal asymptotes because the limits as x approaches infinity do not converge to a finite value. Through calculating these limits and identifying non-existence, we learn more about the variety of end behaviors a function can exhibit. The function in question illustrates that not having a horizontal asymptote does not imply erratic behavior at infinity but signals that the function's value increases or decreases without bound in such scenarios.