- Topic
- Limits
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Limits practice question 14
Evaluate the one-sided limit.
The limit is from the left of 3 only. Which formula does follow on that side?
For the inside is negative, so .
Factor and cancel the common factor.
A fraction, or a decimal to two decimal places.
Theory UBC Math 100 §2.1CLP-1 §1.3Key results
Worked solution
Strategy
On one side of 3 the absolute value has a single formula. Replace it, factor the denominator, cancel x − 3, and substitute.
Working
Left of 3 the inside is negative, so the bars change its sign.
Factor the difference of squares and cancel x − 3.
Substitute into what is left.
Answer
The answer is negative one sixth.
Check
At x = 2.99 the quotient is 0.01/(−0.0599) ≈ −0.167, negative because x² − 9 < 0 just left of 3.
Where this goes wrong
Dropping the bars as if |x − 3| = x − 3, which gives +1/6. That formula holds on the right; from the left the absolute value is 3 − x.
Leans on
Absolute value as a piecewise function, from Functions, graphs and inverses
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