(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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*).

2. Relevant 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*).

3. 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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# Homework Help: Passing the limit through the derivative of a differentiable function

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