End behavior describes what a function does far to the left and far to the right on its graph. Instead of focusing on local details like intercepts or turning points, it asks what happens as x grows without bound in the positive or negative direction. This idea matters because it helps you sketch graphs, compare functions, and understand long term trends in models.
Limits at infinity give precise language for describing this behavior.
Key Facts
- Limit at positive infinity: lim as x -> infinity f(x) describes the y-value f(x) approaches as x becomes very large.
- Limit at negative infinity: lim as x -> -infinity f(x) describes the y-value f(x) approaches as x becomes very negative.
- For polynomials, the leading term controls end behavior: f(x) = a_n x^n + lower degree terms behaves like a_n x^n for large |x|.
- Even-degree polynomial: if a_n > 0, both ends rise; if a_n < 0, both ends fall.
- Odd-degree polynomial: if a_n > 0, left end falls and right end rises; if a_n < 0, left end rises and right end falls.
- For rational functions, compare degrees: if deg numerator < deg denominator, y = 0 is a horizontal asymptote; if degrees are equal, y = leading coefficient ratio is the horizontal asymptote.
Vocabulary
- End behavior
- End behavior is the way a function's values change as x approaches positive infinity or negative infinity.
- Limit at infinity
- A limit at infinity describes the value a function approaches as x becomes extremely large or extremely negative.
- Leading term
- The leading term of a polynomial is the term with the highest power of x, such as 3x^4 in 3x^4 - 2x + 1.
- Horizontal asymptote
- A horizontal asymptote is a horizontal line y = L that a graph approaches as x goes to positive or negative infinity.
- Rational function
- A rational function is a function that can be written as the ratio of two polynomials.
Common Mistakes to Avoid
- Using the constant term to decide end behavior is wrong because the highest-degree term dominates when |x| is very large.
- Assuming every function has a horizontal asymptote is wrong because many polynomials grow without bound instead of approaching a fixed y-value.
- Treating x -> infinity and x -> -infinity as the same direction is wrong because odd powers and some rational functions can behave differently on the two ends.
- Comparing rational functions by plugging in one large number only is unreliable because end behavior is about the trend as x grows without bound, not a single input value.
Practice Questions
- 1 Determine the end behavior of f(x) = -2x^5 + 7x^2 - 4 as x -> infinity and as x -> -infinity.
- 2 Find the horizontal asymptote, if any, of g(x) = (6x^3 - x + 2)/(2x^3 + 5x^2 - 1).
- 3 Explain why h(x) = x^4 - 100x^2 and p(x) = x^4 have the same end behavior even though their graphs can look different near the origin.