Is $\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0$ True?

[gabel]

Is the following true, if no is there som theory i can studdy?

$\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0$

 

[gopher_p]

Is the following true, if no is there som theory i can studdy?

$\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0$

This is not always true. It is true if $f$ is continuous at $x$ or if the $\lim_{\Delta x \to 0}f(x+\Delta x)$ exists; use the product law for limits. It is true if $f$ is bounded near $x$; use the Squeeze Theorem.

If $f$ is unbounded near $x$, the the limit may exist but not be $0$, or it just might fail to exist. Look at $f(x)=1/x$ and $f(x)=1/x^2$ with $x=0$.

 

[gabel]

Thanks, but i really need to show the following if its possibole.

$\lim_{\Delta x \to 0} f(x_0 + \Delta x) \cdot \Delta x= k$ Where k, is a constant.

Is there something i can say aboute f?

 

[HallsofIvy]

What do you mean by "possible"? You have already been told that it is NOT true in general. You have also been told that if f is continuous at $x_0$ or if $\lim_{\Delta x\to 0} f(x+ \Delta x)$ exits then it is true.

 

[gabel]

I was told in general, so there must be som functions that does the oppeist?

 

[HallsofIvy]

I was told in general, so there must be some functions that does the oppeist?

Who told you that? gopher_p, in the only other response here, said "this is not always true".

Take f(x)= 1/x, $x_0= 0$.