- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The definition of the derivative practice question 1
Which limit represents for ?
Compare each choice with the definition of piece by piece. What belongs in the numerator, and what in the denominator?
Start from and substitute .
The denominator is the change in the input, , and takes the root of the whole input.
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
Substitute the function and the point into the definition: the numerator is g(9 + h) − g(9), with the root over the whole input, and the denominator is the change in the input, h.
Working
Write the definition with a = 9 before touching the square root.
Evaluate both pieces. The second is a constant.
Assemble. The denominator must be h, matching the variable in the limit.
Answer
The answer is the limit as h → 0 of (√(9 + h) − 3)/h.
Check
At h = 0.01 this quotient is about 0.1666, close to the slope 1/6 that the power rule gives for √x at 9.
Where this goes wrong
Choosing an option whose denominator is 9 or 9 + h rather than h, or one that reads g(9 + h) as √9 + h. The quotient divides the change in g by the change in the input, and the root covers the whole input.
Leans on
Limit notation, from Limits
Try a similar question
Use the limit definition of the derivative to find for .Saved in this browser only.