Problem 48 - Calculus with Concepts in Calculus

Question

Let $$ f(x)=\frac{x|x|}{x^{2}+1} $$ Show that the graph of \(f\) has two horizontal asymptotes, and determine them.

Step-by-Step Solution

Answer

The horizontal asymptotes are \( y = 1 \) and \( y = -1 \).

Step 1: Understanding Horizontal Asymptotes

Horizontal asymptotes refer to the behavior of a function as the variable approaches positive or negative infinity. For a function \( f(x) \), if \( \lim_{{x \to \pm\infty}} f(x) = L \), then \( y = L \) is a horizontal asymptote.

Step 2: Simplifying Expression for Large |x|

Given the function \( f(x) = \frac{x|x|}{x^2 + 1} \), note \( |x| = x \) when \( x > 0 \) and \( |x| = -x \) when \( x < 0 \). Hence, for large \( x \), the function simplifies to \( f(x) = \frac{x^2}{x^2 + 1} \).

Step 3: Finding the Limit as x Approaches Positive Infinity

Evaluate the limit as \( x \to \infty \). The function becomes \( f(x) = \frac{x^2}{x^2 + 1} \). Dividing numerator and denominator by \( x^2 \), we get \( \frac{1}{1 + \frac{1}{x^2}} \). As \( x \to \infty \), \( \frac{1}{x^2} \to 0 \), so \( \lim_{{x \to \infty}} f(x) = 1 \). Thus, \( y = 1 \) is a horizontal asymptote.

Step 4: Finding the Limit as x Approaches Negative Infinity

Now evaluate the limit as \( x \to -\infty \). The function is \( f(x) = \frac{-x^2}{x^2 + 1} \) when \( x < 0 \). Dividing numerator and denominator by \( x^2 \), we have \( \frac{-1}{1 + \frac{1}{x^2}} \). As \( x \to -\infty \), \( \frac{1}{x^2} \to 0 \), so \( \lim_{{x \to -\infty}} f(x) = -1 \). Hence, \( y = -1 \) is a horizontal asymptote.

Key Concepts

Understanding Limits

Limits are fundamental in calculus used to describe how a function behaves as it approaches a certain point or infinity. In the context of horizontal asymptotes, we leverage limits to determine the value a function tends towards as the input becomes extremely large or extremely small. When solving for limits at infinity, you're essentially investigating what happens to the value of the function as the variable moves toward positive or negative infinity.

For any function \( f(x) \), the expression \( \lim_{{x \to c}} f(x) = L \) signifies that as \( x \) approaches \( c \), \( f(x) \) approaches \( L \). To solve a problem involving horizontal asymptotes, you check the limits of the function as \( x \to \infty \) and \( x \to -\infty \).

Key points to remember about limits include:

• Determining \( \lim_{{x \to \infty}} f(x) \) and \( \lim_{{x \to -\infty}} f(x) \) helps find the horizontal asymptotes.
• The function can approach different values at \(+\infty\) and \(-\infty\), implying potentially different horizontal asymptotes.
• A horizontal asymptote exists at \( y = L \) if \( \lim_{{x \to \pm\infty}} f(x) = L \).

Exploring Infinite Limits

Infinite limits focus on how a function behaves as it approaches very large values or extends into very large negative values. For instance, the expression \( \lim_{{x \to \infty}} f(x) \) or \( \lim_{{x \to -\infty}} f(x) \) evaluates how \( f(x) \) behaves as \( x \) grows larger positively or negatively.

In solving the problem with function \( f(x)=\frac{x|x|}{x^2+1} \), we simplified it for large \( |x| \), determining \( f(x) \approx \frac{x^2}{x^2+1} \) as \( x \to \infty \), and \( f(x) \approx \frac{-x^2}{x^2+1} \) as \( x \to -\infty \). This simplification allows us to compute the limits directly.

Here’s what happens step-by-step:

• As \( x \to \infty \), the contribution of the \( +1 \) in the denominator becomes negligible, making the limit simply \( \lim_{{x \to \infty}} \frac{x^2}{x^2+1} = \frac{1}{1} = 1 \).
• For \( x \to -\infty \), similarly, \( \lim_{{x \to -\infty}} \frac{-x^2}{x^2+1} = -1 \).

Understanding infinite limits helps in predicting and describing the horizontal behavior of functions at extreme values.

Behavior at Infinity

The behavior of a function at infinity tells us about the trends and directions the function follows as it progresses toward larger positive or negative values. This involves an analysis of the horizontal asymptotes.

For the function \( f(x) = \frac{x|x|}{x^2+1} \), we sought to identify horizontal asymptotes by evaluating the function's limits at \( x \to \infty \) and \( x \to -\infty \). The asymptotes indicate where the function settles into a constant behavior.

Here's a summary of how functions behave at infinity:

• An asymptote \( y = L \) means that as \( x \) increases or decreases indefinitely, \( f(x) \) gets closer and closer to \( L \).
• Recognizing the horizontal asymptote informs us about the end behavior, meaning it tells us where the function eventually levels off.
• For \( f(x) = \frac{x|x|}{x^2+1} \), the horizontal asymptotes \( y = 1 \) and \( y = -1 \) denote the function’s leveling behavior at \( +\infty \) and \(-\infty \) respectively.

Understanding behavior at infinity helps in sketching graphs and comprehending long-term trends of functions.