Limits of Rational Functions


In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.


f(x) = (x + 1)/(x² + 3)

Verified step by step guidance

1. Step 1: To find the limit of the rational function \( f(x) = \frac{x + 1}{x^2 + 3} \) as \( x \to \infty \), start by analyzing the degrees of the numerator and the denominator. The numerator \( x + 1 \) is of degree 1, and the denominator \( x^2 + 3 \) is of degree 2.
2. Step 2: Since the degree of the denominator is greater than the degree of the numerator, the limit as \( x \to \infty \) will be determined by the leading terms. Divide each term in the numerator and the denominator by \( x^2 \), the highest power of \( x \) in the denominator.
3. Step 3: After dividing, the expression becomes \( \frac{\frac{x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{3}{x^2}} = \frac{\frac{1}{x} + \frac{1}{x^2}}{1 + \frac{3}{x^2}} \). As \( x \to \infty \), both \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach 0.
4. Step 4: Therefore, the limit of the function as \( x \to \infty \) is \( \frac{0 + 0}{1 + 0} = 0 \).
5. Step 5: To find the limit as \( x \to -\infty \), repeat the same process. The behavior of the function is similar because the terms \( \frac{1}{x} \) and \( \frac{1}{x^2} \) also approach 0 as \( x \to -\infty \). Thus, the limit is \( \frac{0 + 0}{1 + 0} = 0 \).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of rational functions, limits help determine the behavior of the function as the variable approaches infinity or negative infinity, which is crucial for understanding end behavior.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions can vary significantly based on the degrees of the numerator and denominator, especially as x approaches infinity or negative infinity, which influences the limits we calculate.

End Behavior

End behavior refers to the behavior of a function as the input values become very large or very small. For rational functions, this is often determined by the leading terms of the numerator and denominator, which dictate whether the limit approaches a finite value, infinity, or negative infinity.