Rational Function

Definition

A rational function is a function that can be expressed as the quotient of two polynomial functions, typically in the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \neq 0 \). Rational functions are essential in analyzing limits, as they often exhibit behaviors such as vertical asymptotes and holes, which significantly affect the function's limit at specific points.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes at the roots of the denominator where the function is undefined.
2. If the degree of the numerator is less than the degree of the denominator, the limit of the rational function as \( x \) approaches infinity is zero.
3. When both the numerator and denominator have a common factor, this can create a hole in the graph of the function rather than an asymptote.
4. The limit of a rational function can often be determined by simplifying it first, especially when evaluating at points that create indeterminate forms like \( \frac{0}{0} \).
5. Rational functions may also have horizontal asymptotes, which are determined by comparing the degrees of the polynomials in the numerator and denominator.

Review Questions

• How do you determine the limits of a rational function at points where it is undefined?
  • To determine limits at points where a rational function is undefined, you should first factor both the numerator and denominator to identify any common factors. If there are common factors, you can cancel them out and evaluate the limit again to see if it resolves an indeterminate form like \( \frac{0}{0} \). If no common factors exist, you may need to use direct substitution or consider approaching from left and right to analyze behavior near vertical asymptotes.
• In what scenarios would you encounter a hole in the graph of a rational function, and how does it relate to limits?
  • A hole in the graph of a rational function occurs when both the numerator and denominator share a common factor. This situation leads to an indeterminate form when directly substituting the value into the function. When evaluating limits near this point, simplifying the rational function by canceling out the common factor allows for proper evaluation of limits, which reveals that although there is no defined value at that point, limits can still approach specific values.
• Evaluate how understanding rational functions and their limits can impact real-world applications, such as in physics or engineering.
  • Understanding rational functions and their limits is crucial in fields like physics and engineering because these functions often model real-world phenomena such as rates of change and systems approaching equilibrium. For instance, analyzing how velocity changes over time can be represented by a rational function where limits help predict behaviors as time approaches certain critical values. This analysis allows engineers to design more efficient systems by anticipating potential issues like stability or failure points based on limit behaviors in their mathematical models.

Related terms

Polynomial Function: A function represented by a polynomial expression, which consists of variables raised to whole number exponents combined using addition, subtraction, and multiplication.

Vertical Asymptote: A vertical line \( x = a \) where a rational function approaches infinity or negative infinity as it approaches \( a \), indicating that the function is undefined at that point.

Limit: A fundamental concept in calculus that describes the value that a function approaches as the input approaches a certain point.