- Topic
- Limits
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Limits practice question 4
Evaluate the one-sided limit.
Look at the top and the bottom separately as comes down towards 4 from above. What does each approach, and with what sign?
Substituting gives , so the size grows without bound. Decide the sign of for just above 4, such as .
Theory UBC Math 100 §2.1CLP-1 §1.3Key results
Worked solution
Strategy
A non-zero numerator over a denominator tending to 0 means the quotient grows without bound in size. The limit is then +∞ or −∞, and the signs of the top and the bottom on the given side decide which.
Working
Just right of 4, the denominator 4 − x is small and negative.
A positive number over a small negative one is large and negative.
The values decrease without bound, so the limit is −∞.
Answer
The answer is negative infinity.
Check
At x = 4.01 the quotient is 6.01/(−0.01) = −601, and at x = 4.001 it is −6001.
Where this goes wrong
Answering +∞ by reading 4 − x as positive near 4. To the right of 4 it is negative, so the quotient heads to −∞.
Leans on
Sign analysis near a root, from Functions, graphs and inverses
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