Problem 284 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=\frac{1}{x}-\sqrt{x} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 0 \); no horizontal asymptote.

Step 1: Identify Horizontal Asymptotes

To find the horizontal asymptote for the function \( f(x) = \frac{1}{x} - \sqrt{x} \), we evaluate the limit of the function as \( x \to \infty \) and \( x \to -\infty \). As \( x \to \infty \), \( \frac{1}{x} \to 0 \) and \( \sqrt{x} \to \infty \), making the function approach \( -\infty \). Therefore, there is no horizontal asymptote as \( x \to \infty \). As \( x \to -\infty \), \( \sqrt{x} \) is not defined for negative \( x \) in the context of real numbers, so we do not consider this case for a horizontal asymptote.

Step 2: Identify Vertical Asymptotes

To find vertical asymptotes, observe where the function \( f(x) = \frac{1}{x} - \sqrt{x} \) is undefined or where the limits lead to infinity as \( x \to c \). The term \( \frac{1}{x} \) suggests a vertical asymptote at \( x = 0 \), since \( \frac{1}{x} \to \infty \) as \( x \to 0 \). \( \sqrt{x} \) is undefined for negative \( x \), but doesn't affect vertical asymptotes within its defined domain.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are lines that a function approaches as the input, \( x \), moves towards positive or negative infinity. To determine these asymptotes for a given function, one needs to evaluate the limits as \( x \to \infty \) and \( x \to -\infty \).

• If the function approaches a finite number \( L \), then the line \( y = L \) is a horizontal asymptote.
• If the function goes to infinity, there is no horizontal asymptote in that direction.

For the function \( f(x)=\frac{1}{x}-\sqrt{x} \), as \( x \to \infty \), it goes to \(-\infty\) because the term \( -\sqrt{x} \) dominates \( \frac{1}{x} \). Because the function does not approach a finite value, it does not have a horizontal asymptote in this direction. For \( x \to -\infty \), the square root term is not defined for real numbers, so we skip considering this case. This highlights the importance of recognizing how individual terms in a rational function behave as \( x \) grows large.

Vertical Asymptotes

Vertical asymptotes occur at points where a function grows without bound as \( x \) approaches a particular value from the left or the right. These asymptotes often happen where the function becomes undefined.

• Commonly, vertical asymptotes occur where denominators equal zero in rational functions.
• The behavior of the function on both sides of the point needs to be considered.

For the function \( f(x)=\frac{1}{x}-\sqrt{x} \), the term \( \frac{1}{x} \) suggests a vertical asymptote at \( x = 0 \), as \( \frac{1}{x} \) tends to infinity when \( x \to 0 \). The square root \( \sqrt{x} \), being only defined for non-negative \( x \), does not impact vertical asymptotes where it is undefined (negative \( x \)); hence, the only vertical asymptote is at \( x = 0 \). This helps clarify the linking of undefined points and asymptotic behavior in rational functions.

Rational Functions

Rational functions are mathematical expressions formed by dividing two polynomials. They are represented generally as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

• Rational functions can have both horizontal and vertical asymptotes.
• They are continuous except where \( Q(x) = 0 \).

In our given function \( f(x)=\frac{1}{x}-\sqrt{x} \), although it has a rational part \( \frac{1}{x} \), it is not purely rational due to the \( \sqrt{x} \) term. This blend of terms shows how rational functions can merge with other function types, impacting their asymptotic behavior. Rational functions often help illustrate domain restrictions and asymptotic properties clearly, serving as an introduction to more complex function behavior.

Limits

Limits describe the behavior of a function as the input, \( x \), approaches a particular value. They form the foundation for understanding asymptotic behavior and continuity.

• A key use of limits is to find asymptotes by determining how a function behaves as \( x \to \infty \) or \( x \to c \).
• Limits can show both finite values or infinity indicating asymptotic behavior.

In the function \( f(x)=\frac{1}{x}-\sqrt{x} \), limits help us conclude that there is no horizontal asymptote at \( x \to \infty \), as the function approaches \(-\infty\). A similar application of limits confirms the vertical asymptote at \( x = 0 \) as \( \frac{1}{x} \to \infty \). Grasping limits is essential in calculus, providing a powerful tool for analyzing functions.

Domain Restrictions

Domain restrictions define where functions are valid or defined. These restrictions are vital because they affect where we can find asymptotes and analyze functions.

• For rational functions, consider points where the denominator is zero.
• Consider logical constraints like non-negative inputs for square roots.

In the context of \( f(x)=\frac{1}{x}-\sqrt{x} \), the domain is restricted:* \( \frac{1}{x} \) introduces a restriction at \( x = 0 \).* \( \sqrt{x} \) is only defined for non-negative \( x \).Thus, the domain of the function is \( x > 0 \). Recognizing and adhering to domain restrictions is fundamental to successfully analyzing a function's behavior, allowing for a complete understanding of its properties and limitations.