Limits at Infinity and Asymptotes

What happens at the far ends of a graph.

12 problems · about 21 minutes · LIM-2.D

What this lesson teaches

The student evaluates limits at infinity and identifies horizontal and vertical asymptotes.

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.

Straightforward practice. Get the method working first.

5 problems

Limit of (8x + 3) ÷ (4x − 1) as x grows without bound?
Answer 2
Why 2.
A denominator is zero and the numerator is not. What is at that point?
Answer A vertical asymptote.
Why A vertical asymptote.
Limit of (4x) ÷ (x + 1) as x grows without bound?
Answer 4
Why 4 ÷ 1.
Limit of (x + 2) ÷ (x² + 1) as x grows without bound?
Answer 0
Why The bottom grows faster.
Limit of (9x² + x) ÷ (3x² − 4) as x grows without bound?
Answer 3
Why 9 ÷ 3.

The same ideas with more to keep track of.

3 problems

f(x) = 5 ÷ (x + 2). Where is the vertical asymptote?
Answer -2
Why x + 2 = 0.
f(x) = 1 ÷ x. What is the horizontal asymptote value?
Answer 0
Why The bottom grows without bound.
f(x) = (x² − 4) ÷ (x − 2). Is x = 2 an asymptote or a hole? 1 for asymptote, 2 for hole.
Answer 2
Why The factor cancels.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Which produce a horizontal asymptote of y = 0?
Answer Degree 1 over degree 2; Degree 2 over degree 5
Why A larger bottom degree always wins.
Limit of (7x³) ÷ (2x³ + 9) as x grows without bound, as a decimal?
Answer 3.5
Why 7 ÷ 2.
The Far End: Limit of (6x + 1) ÷ (2x − 5) as x grows without bound?
Answer 3
Why 3.
The Break: f(x) = 1 ÷ (x − 7). At which x is the vertical asymptote?
Answer 7
Why x = 7.