Difference Quotient : Limits and Differentiable Functions
Not what you're looking for?
Assume f:(-1,1) --> R and f'(0) exists.
If a_n , b_n -> 0 as n->infinity, define the difference quotient:
D_n = ( f(b_n) - f(a_n) ) / ( b_n - a_n).
a) Prove lim [n -> infinity] D_n = f'(0) under each condition below:
(i) a_n < 0 < b_n .
(ii) 0 < a_n < b_n and (b_n) / (b_n - a_n) <= M
(iii) f'(x) exists and is continuous forall x in (-1,1).
b) Set f(x) = (x^2)sin(1/x) for x neq 0 and f(0)=0.
Observe that f is diff everywhere in (-1,1) and f'(0)=0.
Find a_n, b_n that tend to 0 in such a way that D_n converges to a limit not equal to f'(0).
Purchase this Solution
Solution Summary
Difference quotients, limits and differentiable functions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Since exists and as , then we have
and .
So for any , we can find some , such ...
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.