Practice question · Multiple choice
The derivative is defined as a limit of 0/0, the difference quotient is undefined at h = 0 for every function. Why is that not a fatal problem?
Hints
- The limit lesson insisted on ignoring the value AT the point. Ask why that mattered.
- At h = 0 there is no line through one point. What does the limit supply instead?
Show the answer
B. Because a limit deliberately excludes the point itself
Why
This is what limits were for. The quotient is 0/0 at h = 0 for every function ever differentiated, so a definition requiring the point would define nothing, the limit describes the approach instead. Newton and Leibniz used the idea for a century before Cauchy and Weierstrass made it rigorous.
Practise The Derivative
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