- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Curve sketching practice question 4
The graph of has one horizontal asymptote, the line . Find .
When is huge, which term on the top and which on the bottom matter most?
A horizontal asymptote is the limit of as . Divide every term by , the highest power, and let grow.
A number, as a fraction or a decimal.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
A horizontal asymptote is where f settles as x → ∞ or x → −∞. Divide the top and the bottom by the highest power of x, here x², and let the other terms vanish.
Working
Divide every term by x².
1/x and 7/x² tend to 0 in both directions, so one line serves both ends.
Answer
The answer is y equals five halves.
Check
At x = 100, f(x) = 49,900/20,007 ≈ 2.49, already close to 2.5.
Where this goes wrong
Answering 0, or 2/5. When the top and the bottom have the same degree, the asymptote is the ratio of their leading coefficients, top over bottom: 5/2.
Leans on
Limits at infinity, from Limits
Try a similar question
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