A limit at infinity asks what a function settles toward as x runs far out, rather than as it approaches a specific number.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Horizontal asymptotes
If the function approaches a fixed value L far out, the line y = L is a horizontal asymptote.
The degree rule
For a rational function: equal degrees give the ratio of leading coefficients, a smaller top gives 0, and a larger top gives no horizontal asymptote.
They may be crossed
A horizontal asymptote describes long-run behaviour only. A curve may cross it near the origin and still approach it far out.
Vertical asymptotes
These occur where the denominator is zero and the numerator is not. The values run away, so the limit there does not exist.
Asymptote or hole
If top and bottom share the zero, the graph has a hole rather than an asymptote. Factor before deciding.
Limits at infinity describe the far ends
A limit as x tends to infinity says what the function settles towards as the input grows without bound. A finite answer means a horizontal asymptote; the limit describes long-run behaviour rather than a point.
Divide by the highest power
For a rational function, divide numerator and denominator by the highest power of x in the denominator. Every term with x underneath then tends to zero, and the surviving terms give the answer.
The shortcut from the degrees
Numerator degree lower gives limit 0; equal gives the ratio of leading coefficients; higher gives no finite limit. The shortcut follows from the division technique and saves doing it every time.
Infinite limits are a different statement
A limit at infinity has infinity as the input; an infinite limit has it as the output and signals a vertical asymptote. The two phrases look similar and describe opposite features, so read them carefully.
Step 2: Try It Yourself
Tap and try it out.
- Point(3, 0.67)
Step 3: Watch an Example
One step at a time.
Watch Marcus Find Both Asymptotes
Marcus is given f(x) = (2x + 1) ÷ (x − 3).
- Step 1
For the vertical asymptote he sets the denominator to zero, giving x = 3.
Step 4: Your Turn
Practice makes it stick.
The Far End
Problem 1 of 2
Limit of (6x + 1) ÷ (2x − 5) as x grows without bound?
The Break
Problem 2 of 2
f(x) = 1 ÷ (x − 7). At which x is the vertical asymptote?
Ends and Breaks
1 of 8
Limit of (4x) ÷ (x + 1) as x grows without bound?
2 of 8
Limit of (x + 2) ÷ (x² + 1) as x grows without bound?
3 of 8
Limit of (9x² + x) ÷ (3x² − 4) as x grows without bound?
4 of 8
f(x) = 5 ÷ (x + 2). Where is the vertical asymptote?
5 of 8
f(x) = 1 ÷ x. What is the horizontal asymptote value?
6 of 8
f(x) = (x² − 4) ÷ (x − 2). Is x = 2 an asymptote or a hole? 1 for asymptote, 2 for hole.
7 of 8
Which produce a horizontal asymptote of y = 0?
8 of 8
Limit of (7x³) ÷ (2x³ + 9) as x grows without bound, as a decimal?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of (8x + 3) ÷ (4x − 1) as x grows without bound?
Question 2 of 2
A denominator is zero and the numerator is not. What is at that point?
What You Learned
- A limit at infinity gives the horizontal asymptote.
- Compare degrees: equal gives a ratio, smaller top gives 0, larger top gives none.
- A vertical asymptote needs a zero denominator and a non-zero numerator.