Infinite limits describe what happens when a function grows without bound as x gets close to a certain value. Instead of approaching a regular number, the function values rise toward positive infinity or fall toward negative infinity. This idea matters because it explains graph behavior near breaks, undefined inputs, and extreme rates of change.
A vertical asymptote is the graph feature that often marks this kind of unbounded behavior.
Key Facts
- lim x -> a f(x) = infinity means f(x) becomes arbitrarily large and positive as x approaches a.
- lim x -> a f(x) = -infinity means f(x) becomes arbitrarily large and negative as x approaches a.
- A vertical asymptote occurs at x = a if lim x -> a+ f(x) = infinity, lim x -> a+ f(x) = -infinity, lim x -> a- f(x) = infinity, or lim x -> a- f(x) = -infinity.
- One-sided limits can differ: lim x -> a- f(x) and lim x -> a+ f(x) describe behavior from the left and right of x = a.
- For rational functions, vertical asymptotes often occur where the denominator equals 0 after canceling any common factors.
- Example: f(x) = 1/(x - 2) has a vertical asymptote at x = 2, with lim x -> 2- f(x) = -infinity and lim x -> 2+ f(x) = infinity.
Vocabulary
- Infinite limit
- An infinite limit occurs when function values increase or decrease without bound as x approaches a specific value.
- Vertical asymptote
- A vertical asymptote is a vertical line x = a that a graph approaches while the function values grow without bound.
- One-sided limit
- A one-sided limit describes the value or behavior of a function as x approaches a point from only the left or only the right.
- Rational function
- A rational function is a function that can be written as a ratio of two polynomials.
- Unbounded behavior
- Unbounded behavior means the outputs of a function do not stay within any finite range near a point or over an interval.
Common Mistakes to Avoid
- Calling infinity a number, which is wrong because an infinite limit describes unbounded behavior rather than a finite limit value.
- Ignoring one-sided limits, which is wrong because the left and right sides of a vertical asymptote can go to different infinities or one side may not be defined.
- Assuming every zero of a denominator is a vertical asymptote, which is wrong because a common factor may cancel and create a removable hole instead.
- Writing x = infinity as a vertical asymptote, which is wrong because vertical asymptotes are vertical lines with equations like x = a.
Practice Questions
- 1 Find the vertical asymptote and the two one-sided infinite limits for f(x) = 3/(x - 4).
- 2 For g(x) = (x + 1)/((x - 2)(x + 3)), find all vertical asymptotes and state whether g(x) goes to infinity or -infinity from the right side of each asymptote.
- 3 Explain why h(x) = (x - 5)/((x - 5)(x + 2)) does not have a vertical asymptote at x = 5, and describe what happens there instead.