- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
The definition of the derivative practice question 23
A function is differentiable, with and . Evaluate the limit below.
Compare this quotient with the definition of . Which two inputs does the definition compare, and which two does this quotient compare?
Subtract and add in the numerator, and split the quotient into two: one with , one with .
Rescale each piece to match the definition: , and as .
A number.
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
The limit is not the definition as written, because the two inputs are 3 + 2h and 3 − 3h. Insert f(3), split into two quotients, and rescale each into the form (f(3 + k) − f(3))/k with k → 0.
Working
Subtract and add f(3), which changes nothing.
Make each denominator the change in the input: 2h for the first piece, −3h for the second.
Each rescaled piece is a difference quotient at 3 whose step tends to 0.
The value f(3) = 5 cancelled in the first step; only the slope matters.
Answer
The answer is negative 10.
Check
For f(x) = 11 − 2x, which has f(3) = 5 and f′(3) = −2, f(3 + 2h) − f(3 − 3h) = −10h exactly, so the limit is −10.
Where this goes wrong
Reading any quotient with an h underneath as f′(3) and answering −2. The inputs 3 + 2h and 3 − 3h are 5h apart, so the quotient is about 5 times the slope.
Try a similar question
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