How can I prove the limit of a quotient using the definition of derivative?

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In summary, using the definition of derivative, we can show that the limit of f(z)/g(z) as z approaches z(0) is equal to f'(z(0))/g'(z(0)) when f(z(0))=g(z(0))=0 and both f'(z(0)) and g'(z(0)) exist, with g'(z(0)) not equal to 0. By simplifying f'(z(0)) and g'(z(0)), we can see that they both have a common factor of (z-z(0)), allowing us to cancel it out and show that the limit is indeed f'(z(0))/g'(z(0)).
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Homework Statement



Supposed f(z(0))=g(z(0))=0 and that f'(z(0)) and g'(z(0)) exist where g'(z(0)) is not equal to 0. Use definition (10, section 19 of derivative to show that :

lim z->z(0) (f(z)/g(z))=f'(z(0))/g'(z(0))


Homework Equations



definition 1: f'(z(0))=lim z->z(0) f(z)-f(z(0))/(z-z(0)

The Attempt at a Solution



f'(z(0))= lim (z->z(0)) f(z)-f(z(0))/(z-z(0))=f(z)-0/(z-z(0))=f(z)/(z-z(0)); Not sure where I should with with this proof. Help.
 
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You started off really good, except you forgot the carry the limit operator through, which is important. Now that you've simplified f'(z(0)), do the same for g'(z(0)) ! Remember to use all the information you're given.
 

1. What is the definition of a derivative?

The derivative of a function at a certain point is the limit of the slope of the function at that point as the change in the input approaches zero.

2. How can the limit of a quotient be used to prove the derivative?

The limit of a quotient can be used to prove the derivative by rewriting the definition of the derivative as a quotient of two small changes, and then taking the limit as those changes approach zero.

3. What is the significance of using the definition of derivative to prove a limit of a quotient?

Using the definition of derivative to prove a limit of a quotient allows for a more rigorous and precise understanding of the concept. It also allows for the proof to be applied to a wider range of functions, rather than relying on specific derivative rules.

4. Can the definition of derivative be applied to all types of functions?

Yes, the definition of derivative can be applied to all types of functions, including polynomial, exponential, and trigonometric functions. However, the process of proving the derivative using the definition may vary depending on the type of function.

5. Is using the definition of derivative the only way to prove a limit of a quotient?

No, there are other methods for proving a limit of a quotient, such as using derivative rules or the mean value theorem. However, using the definition of derivative is considered the most rigorous and generalizable approach.

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