Passing the limit through the derivative of a differentiable function

In summary, the homework statement asks if there exists a sequence of numbers that converges to a specific point and has a limit equal to the given function. The attempt at a solution states that since f is differentiable on an open interval, its derivative is continuous on this interval. The conclusion is that there exists a sequence of numbers that converges to the given point and has a limit equal to the given function.
  • #1
kbgregory
7
0

Homework Statement



Suppose f is differentiable on an open interval I and let x* [tex]\in[/tex] I. Show that there exists a sequence {x_n}[tex]\subset[/tex] I such that lim[n->inf]{x_n}=x* and lim[n->inf]{f'(x_n)}=f'(x*).


Homework Equations



We know that a function g is continuous iff for any sequence {x_n} with lim[n->inf]{x_n}=x*, lim[n->inf]{g(x_n)}=g(x*).

The Attempt at a Solution



I think I need to show that since f is differentiable on I, then its derivative is continuous on I, and since its derivative is continuous on I, then there exists a sequence {x_n} with lim[n->inf]{x_n}=x* for which lim[n->inf]{f'(x_n)}=f'(x*).

But I am not sure how to show this, or even if its right.
 
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  • #2
No, f differentiable on I does not imply its derivative is continuous. The canonical counterexample is f(x) = (x^2)*sin(1/x) if x =/= 0 and f(x) = 0 if x = 0.

In the conclusion, you have a double limit. There is a certain procedure to apply to these limits that would make the problem very simple. The continuity of f would be essential in this case.

But I think it's probably more illuminating to work from first principles. Write out the definition of the derivative of f at x* (epsilon-delta definition of limit), pick a specific sequence converging to x*, and make the appropriate modifications to your definition to show the conclusion.
 
  • #3
I did what snipez90 said, but I got stuck. I know that it is necessary to apply the Mean Value Theorem, but I don't really understand how. Maybe you can figure out how the MVT can be applied.
 
  • #4
Take your [tex] x \in I [/tex]. You know that there is a sequence of positive numbers, [tex] h_n [/tex] such that [tex] \lim_{n \rightarrow \infty} h_n = 0 [/tex]. Thus [tex] \lim_{n \rightarrow \infty} \frac{f(x + h_n) - f(x)}{h_n} = f'(x) [/tex].

Now look at the difference quotient [f(x + h_n) - f(x)] / h_n . By the MVT [f(x + h_n) - f(x)] / h_n = f'(c_n) for x < c_n < x + h_n

As n goes to infinity that difference quotient gets closer and closer to the derivative of f at x, right? So what can you say about the sequence f'(c_n)?
 

What does it mean to "pass the limit through the derivative" of a differentiable function?

Passing the limit through the derivative of a differentiable function refers to taking the limit of the derivative of a function as the independent variable approaches a given value. In other words, it involves finding the derivative of a function at a particular point by taking the limit of the difference quotient as the change in the independent variable approaches 0.

Why is it important to understand how to pass the limit through the derivative of a differentiable function?

Understanding how to pass the limit through the derivative of a differentiable function is crucial in calculus and other areas of mathematics. It allows us to find the slope of a tangent line to a curve at a given point, which is essential in optimization, related rates, and curve sketching problems.

What is the difference between finding the derivative of a function and passing the limit through the derivative?

Finding the derivative of a function involves using the rules of differentiation to find a general formula for the derivative. On the other hand, passing the limit through the derivative involves evaluating the derivative at a specific point by taking the limit of the difference quotient. In other words, finding the derivative gives us a general formula, while passing the limit through the derivative gives us a specific value.

Can every function be differentiated using the limit definition of the derivative?

Yes, every function that is differentiable can be differentiated using the limit definition of the derivative. This is because the limit definition of the derivative is a general method for finding the derivative of a function, and it applies to all differentiable functions.

What are some common mistakes when passing the limit through the derivative of a differentiable function?

Some common mistakes when passing the limit through the derivative include not simplifying the difference quotient before taking the limit, forgetting to include the limit notation, and incorrectly applying the rules of limits. It is also important to remember to take the limit as the change in the independent variable approaches 0, not the actual value of the independent variable.

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