- Topic
- Limits
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Limits practice question 10
Evaluate the limit.
Far out along the x-axis, only the highest-degree term in each of the numerator and denominator still matters.
Divide every term on the top and bottom by , then take the limit.
A decimal or a fraction.
Theory UBC Math 100 §2.1.2CLP-1 §1.5Key results
Worked solution
Strategy
For a rational function as x grows without bound, divide numerator and denominator by the highest power of x present. Every lower-order term then vanishes.
Working
Divide every term above and below by x². Nothing is lost; this is one fraction rewritten.
Each remaining reciprocal term shrinks to zero as x grows.
What survives is the ratio of the leading coefficients, sign included.
Answer
The answer is −3/2, or −1.5.
Check
At x = 1000 the quotient is 2,995,001/(−1,999,993) ≈ −1.4975, negative and close to −1.5.
Where this goes wrong
Reporting 3/2 and losing the sign, because the denominator is written as 7 − 2x² rather than −2x² + 7. The leading coefficient of the denominator is −2.
Leans on
Behaviour of 1/x as x grows, from Functions, graphs and inverses
Saved in this browser only.