Differentiation rules practice question 14
Evaluate the limit by recognising it as a derivative.
Compare with the definition . What are and ?
Once you have and , the power rule does the rest.
A whole number.
Theory UBC Math 100 §4.1CLP-1 §2.6Key results
Worked solution
Strategy
Factoring x7 − 37 is possible but long. The quotient is exactly the difference quotient of x7 at 3, so the limit is a derivative, and the power rule evaluates it in one line.
Working
Match the pattern of the definition of f′(3).
Power rule at x = 3.
Answer
The answer is 5103.
Check
At x = 3.001 the quotient is about 5108, and at x = 2.999 about 5098, either side of 5103.
Where this goes wrong
Substituting x = 3 and concluding the limit is 0/0 hence undefined, or attempting a seven-term factorisation. The limit has the shape of a derivative, and that is the fast route.
The idea behind it
The differentiation rules lesson, a figure to move, in a few steps
Leans on
The limit definition of the derivative, from The definition of the derivative
Try a similar question
Use the limit definition of the derivative to find for .Saved in this browser only.