- Topic
- Limits
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Limits practice question 7
Find the horizontal asymptote of the graph of the function below: enter the value of it approaches as .
For huge , only the leading terms of the top and the bottom matter. The top is a product of two linear factors: what degree is it, and what is its leading term?
Expand the top only as far as its term, which is .
Divide every term on the top and the bottom by , then let .
An exact value, or a decimal to three places.
Theory UBC Math 100 §2.2CLP-1 §1.5Key results
Worked solution
Strategy
A horizontal asymptote is the limit of f as x → ∞ or x → −∞. The top and the bottom are both quadratics, so divide each by x²: only the x² terms survive, and they give the same value at both ends.
Working
Expand the top; its x² term is −4x², sign included.
Divide every term above and below by x².
Each term with x in a denominator tends to 0, whichever way x goes, so both ends give the line y = −2/3.
Answer
The answer is y = −2/3, about −0.667.
Check
f(1000) ≈ −0.6685 and f(−1000) ≈ −0.6648, both close to −2/3 ≈ −0.6667.
Where this goes wrong
Losing the minus sign of −4x and answering 2/3. Far from the origin the top behaves like −4x², which is negative for large x of either sign, while the bottom is positive, so the asymptote lies below the x-axis.
Leans on
Dominant terms for large x, from Power functions and first sketches
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