- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
The definition of the derivative practice question 25
Let be the function below. Which statement about the one-sided limit is correct?
For , which formula gives ? And which formula gives ?
, but . What does the numerator tend to, and what happens when a number near is divided by a small negative ?
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
Write out the quotient with the right formula for each value: f(2) from the piece that holds at x = 2, and f(2 + h) from the piece for x < 2. If f jumps at 2, the top does not tend to 0 and the quotient cannot settle.
Working
f(2) comes from the second formula, which holds for x ≥ 2; f(2 + h) for h < 0 comes from the first.
Expand, then divide each term by h.
For small negative h, −1/h is large and positive.
f jumps at 2, so it is not continuous there from the left, and it has no derivative from the left.
Answer
The answer is it tends to positive infinity.
Check
At h = −0.01 the quotient is (1.99² − 5)/(−0.01) = (3.9601 − 5)/(−0.01) = 103.99, and at h = −0.001 it is 1003.999.
Where this goes wrong
Differentiating the left-hand formula and reporting 4. The definition subtracts f(2), which here comes from the other formula, so a jump in f makes the left-hand quotient grow without bound.
Leans on
One-sided limits, from Limits
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