- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
The definition of the derivative practice question 15
Use to evaluate the limit below.
The prompt hands you the limit of . How could you make that exact shape appear inside this quotient?
Split the exponential: . What common factor does the numerator then have?
, and does not depend on .
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §3.5Key results
Worked solution
Strategy
The quotient is the difference quotient of eˣ at x = 2, so the limit is the slope of eˣ there. Factoring out e² reduces it to the limit that defines e.
Working
Exponent law, then factor out the constant e².
e² does not depend on h, so it comes out of the limit; the limit that is left is 1.
The limit is the slope of eˣ at 2. The same working at any point shows that eˣ is its own derivative.
Answer
The answer is e squared, about 7.389.
Check
At h = 0.001 the quotient (e2.001 − e²)/0.001 is about 7.393, close to e² ≈ 7.389.
Where this goes wrong
Substituting h = 0 and stopping at 0/0. The form says only that the quotient needs rewriting: factor out e² and the limit that defines e finishes it.
Leans on
Laws of exponents, from Functions, graphs and inverses
Try a similar question
For a base , the slope of at is the limit below.Saved in this browser only.