Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.


f(x)=|1−x^2| / x(x+1)

Verified step by step guidance

1. The function given is \( f(x) = \frac{|1-x^2|}{x(x+1)} \). We need to analyze the behavior of this function as \( x \to \infty \) and \( x \to -\infty \).
2. For large values of \( x \), the term \( x^2 \) dominates \( 1 \) in \( |1-x^2| \), so \( |1-x^2| \approx |x^2| = x^2 \). Thus, \( f(x) \approx \frac{x^2}{x(x+1)} = \frac{x^2}{x^2 + x} \).
3. Simplify \( \frac{x^2}{x^2 + x} \) by dividing the numerator and the denominator by \( x^2 \), resulting in \( \frac{1}{1 + \frac{1}{x}} \). As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( \lim_{x \to \infty} f(x) = \frac{1}{1+0} = 1 \).
4. For \( x \to -\infty \), the simplification is similar: \( f(x) \approx \frac{x^2}{x^2 + x} = \frac{1}{1 + \frac{1}{x}} \). As \( x \to -\infty \), \( \frac{1}{x} \to 0 \), so \( \lim_{x \to -\infty} f(x) = 1 \).
5. Since both \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \) are equal to 1, the horizontal asymptote of the function is \( y = 1 \).

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine how the function behaves in extreme cases, which is crucial for identifying horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity exists and is a finite number, it indicates a horizontal asymptote at that value.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They represent the limiting value of a function in these extreme cases. To find horizontal asymptotes, one typically evaluates the limits of the function at both ends of the x-axis, determining if the function stabilizes at a particular value.

Absolute Value Functions

Absolute value functions, such as |1 - x^2|, affect the behavior of a function by ensuring that outputs are non-negative. This can change the function's limits and asymptotic behavior, especially when the expression inside the absolute value changes sign. Understanding how to handle absolute values is essential for accurately analyzing limits and identifying asymptotes.