- Topic
- Limits
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Limits practice question 20
Find the horizontal asymptote of the graph of the function below as : enter the value of it approaches.
On the far left is negative. Before any algebra, is positive or negative there, and how does compare with when ?
Take out of the root: . What is when is negative?
Check the sign before computing: for very negative , the top is negative and the root is positive.
An exact value, or a decimal to three places.
Theory UBC Math 100 §2.2CLP-1 §1.5Key results
Worked solution
Strategy
Divide the top and the bottom by the dominant power, x. The root hides an absolute value, √(x²) = |x|, and on the far left |x| = −x, which sets the sign.
Working
The square root of x² is |x|, and |x| = −x for negative x.
Divide the top and the bottom by x; the minus sign stays below.
The terms in 1/x and 1/x² vanish, and √9 = 3.
Answer
The answer is y = −4/3, about −1.333.
Check
At x = −100, f = −399/√90,002 ≈ −1.330, negative because the top is negative and the root positive.
Where this goes wrong
Writing √(9x²) = 3x and getting 4/3, which is the asymptote as x → ∞. For negative x, √(9x²) = 3|x| = −3x: the top is negative while the root is positive, so the left end is at −4/3.
Leans on
Square roots and absolute value, from Functions, graphs and inverses
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