MATH1851 Chap.3 Limits at Infinity and Asymptotes
Limits at Infinity and Asymptotes
Separate two meanings of infinity
An infinite limit keeps the input near a finite point and records output growing without bound. A limit at infinity sends the input far along the number line and asks about end behaviour.
This distinction maps directly to graph geometry: the first may create a vertical asymptote, while a finite end limit creates a horizontal asymptote.
Use degree evidence carefully
For rational functions at infinity, leading degrees control the result.
A lower numerator degree gives zero, equal degrees give the ratio of leading coefficients, and a higher numerator degree requires sign analysis or polynomial division. The shortcut addresses end behaviour only. It does not locate holes or vertical asymptotes at finite inputs.
Find finite singularities before end behaviour
Start with denominator zeros, then factor and cancel.
A cancelled factor produces a removable hole rather than unbounded behaviour. At each uncancelled zero, check one-sided signs to determine positive or negative infinity.
Only after local defects are classified should you move to horizontal or oblique end behaviour.
Oblique lines preserve the dominant trend
A line y equals mx plus b is an oblique asymptote when the difference between the function and the line tends to zero. Polynomial division is often the cleanest route: the quotient supplies the dominant line and the remainder over the denominator vanishes.
The curve may cross an asymptote at a finite point because an asymptote describes limiting distance, not a barrier.
Report lines with evidence
A complete answer pairs every vertical line with a one-sided unbounded limit and every horizontal or oblique line with its corresponding end limit.
Listing the evidence beside the line prevents a cancelled hole from being mislabeled and makes the positive and negative end directions explicit.
What this chapter covers
- 01
Infinite limits near a finite input
- 02
Limits at positive and negative infinity
- 03
Rational degree comparison
- 04
One-sided sign charts at denominator zeros
- 05
Horizontal and vertical asymptotes
- 06
Oblique asymptotes through polynomial division
Worked example · free
List every asymptote of one rational function
- 3Test the denominator zero for cancellation and one-sided unbounded behaviour.
- 3Divide the polynomials to identify the dominant end line.
- 3State every asymptote and rule out the inapplicable type.
Key terms
- Infinite Limit
- Output increasing or decreasing without bound as the input approaches a finite target.
- Limit At Infinity
- The behaviour of a function as the input grows without bound in a specified direction.
- Vertical Asymptote
- A line x equals c associated with unbounded behaviour from at least one side of c.
- Horizontal Asymptote
- A line y equals L associated with a finite function limit at positive or negative infinity.
- Oblique Asymptote
- A slanted line whose difference from the function tends to zero along an infinite direction.
- Leading Degree
- The largest polynomial exponent, which determines rational end behaviour together with its coefficient.
Limits at Infinity and Asymptotes FAQ
Does every denominator zero give a vertical asymptote?
No. Factor the numerator and denominator first. If the zero comes from a common factor that cancels, the graph has a removable hole when the simplified limit is finite. An uncancelled zero still needs a one-sided sign check to describe the unbounded behaviour.
Can a graph cross a horizontal asymptote?
Yes. A horizontal asymptote describes what happens as the input tends to positive or negative infinity. It does not forbid finite intersections. The same idea applies to an oblique asymptote: the relevant condition is that the difference tends to zero at the end.
When is polynomial division better than degree comparison?
Use division when the numerator degree is one or more above the denominator degree and you need the dominant polynomial trend. When the degree difference is exactly one, the quotient directly gives the oblique line while the remainder term shows why the difference vanishes.
Exam move
For every rational function, use the same two-column audit. In the local column, list denominator zeros, cancellations and one-sided signs. In the end-behaviour column, compare degrees, coefficients and positive versus negative infinity. Practise radicals separately and replace square root of x squared with absolute x before choosing a sign.
Finish each problem with an explicit list of asymptote equations and the limit statement supporting each line. This closing inventory catches missed vertical or oblique features.
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