What Infinite Limits Tell You
Most limits ask what value a function approaches as gets close to a number. But sometimes the function does not approach a finite number at all. Instead, its values become arbitrarily large or arbitrarily negative.
For example, consider as approaches 2. The denominator gets closer and closer to zero, while the numerator stays positive. The function therefore becomes extremely large in magnitude.
When the function grows without bound, we describe the behavior with an infinite limit. This is directly connected to the graph: the line is a vertical asymptote.
The Connection
If the function increases without bound as approaches , then:
If the function decreases without bound instead, then:
In either case, the vertical line is a vertical asymptote.
Notice an important point: the function does not have to equal infinity. Infinity is not a number that the function reaches. The notation describes a function whose values become larger than any fixed positive number, or smaller than any fixed negative number, as the input gets sufficiently close to the asymptote.
One-Sided Limits Matter
Near a vertical asymptote, the behavior from the left may be different from the behavior from the right. That is why AP Calculus frequently asks for one-sided limits.
These two statements can both be true even though the two-sided limit does not exist. If one side approaches and the other approaches , there is no single infinite behavior shared by both sides.
A Step-by-Step Method
- Find the input value where the denominator becomes zero. For a rational function, this is often the first place to look for a vertical asymptote.
- Factor and simplify the function if possible. A factor that cancels may produce a removable discontinuity rather than a vertical asymptote.
- Check the sign on each side of the critical value. Determine whether the function becomes positive or negative as the denominator approaches zero.
- Evaluate the one-sided limits. Decide whether each side approaches , , or a finite value.
- State the vertical asymptote. If the function approaches or from either side, then the corresponding vertical line is a vertical asymptote.
Worked Examples
Determine the behavior of as approaches 3, and identify any vertical asymptote.
Analyze as approaches 2.
Determine whether has a vertical asymptote at .
Find all vertical asymptotes of .
How to Read a Graph
You can often identify an infinite limit visually. As the graph gets closer to a vertical line, watch what happens to the -values.
- If the graph rises without bound on one side, the corresponding one-sided limit is .
- If the graph falls without bound on one side, the corresponding one-sided limit is .
- If the graph approaches a vertical line with unbounded behavior, that line is a vertical asymptote.
- Always inspect the left and right sides separately before making a statement about a two-sided limit.
Infinite Limit vs. Vertical Asymptote
These ideas are closely related, but they are not exactly the same statement. An infinite limit describes the behavior of the function as the input approaches a particular value. A vertical asymptote is the vertical line associated with that unbounded behavior.
So, if either one-sided limit is or , the corresponding line is a vertical asymptote.
Common Mistakes
Writing is misleading. Infinity is not an ordinary real-number output. Instead, write an infinite limit such as to describe unbounded behavior.
A common factor may cancel. For example, simplifies to , so the discontinuity at is a hole rather than a vertical asymptote.
The left-hand and right-hand limits can have opposite signs. Always check and separately when investigating a vertical asymptote.
A vertical asymptote describes what happens as approaches a finite value. A horizontal asymptote describes what happens as becomes very large or very negative. They answer different limit questions.
Practice Problems
Evaluate the one-sided limits and identify the vertical asymptote for .
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Determine and identify the vertical asymptote for .
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Does have a vertical asymptote at ? Explain.
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Suppose and . What can you conclude about ?
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