Limits at infinity describe what happens to a function as x becomes very large positive or very large negative. Instead of asking for the value at one point, they ask about the long-term trend of the graph. This idea matters because many real systems settle toward a steady value, grow without bound, or oscillate as time or distance increases.
On a graph, a limit at infinity often appears as a curve flattening toward a line.
Key Facts
- lim x->infinity f(x) = L means f(x) gets closer to L as x increases without bound.
- lim x->-infinity f(x) = L means f(x) gets closer to L as x decreases without bound.
- A horizontal asymptote y = L occurs when lim x->infinity f(x) = L or lim x->-infinity f(x) = L.
- For rational functions, compare the degrees of the numerator and denominator to find end behavior.
- If degrees are equal, the horizontal asymptote is y = ratio of leading coefficients.
- If denominator degree is larger, the horizontal asymptote is y = 0.
Vocabulary
- Limit at infinity
- A limit at infinity describes the value a function approaches as x grows without bound in the positive or negative direction.
- Horizontal asymptote
- A horizontal asymptote is a horizontal line that a graph approaches as x goes toward infinity or negative infinity.
- Rational function
- A rational function is a function that can be written as one polynomial divided by another polynomial.
- Leading term
- The leading term is the term with the highest power of x in a polynomial.
- End behavior
- End behavior describes how a function acts as x becomes very large positive or very large negative.
Common Mistakes to Avoid
- Using small x-values to decide a limit at infinity is wrong because limits at infinity depend on long-term behavior, not nearby values.
- Ignoring leading terms in a rational function is wrong because the highest powers determine the end behavior as |x| becomes very large.
- Assuming a horizontal asymptote can never be crossed is wrong because a graph may cross its horizontal asymptote at finite x-values and still approach it in the long run.
- Treating x->infinity and x->-infinity as always the same is wrong because some functions approach different values or grow in different directions on each end.
Practice Questions
- 1 Find lim x->infinity (3x^2 + 5x - 1)/(2x^2 - 7). State the horizontal asymptote.
- 2 Find lim x->-infinity (4x - 9)/(x^2 + 1). State the horizontal asymptote.
- 3 A rational function has numerator degree 3 and denominator degree 2. Explain whether it has a horizontal asymptote and describe its likely end behavior.