Proving Derivative of sin(x)/x is Zero at x=0

[mherna48]

Hey can anyone help me prove that the derivative of sin(x)/x is zero at x=0

 

[snipez90]

A function f:R -> R is differentiable at 0 if
\lim_{h \rightarrow 0}\frac{f(h)-f(0)}{h}
exists. You can apply this definition and then use l'Hopital's rule or perhaps some first-order estimates to show that this last limit exists. Alternatively, you can probably work directly with the series expansion for sin.

 

[evagelos]

A function f:R -> R is differentiable at 0 if
\lim_{h \rightarrow 0}\frac{f(h)-f(0)}{h}
exists. You can apply this definition and then use l'Hopital's rule or perhaps some first-order estimates to show that this last limit exists. Alternatively, you can probably work directly with the series expansion for sin.

`
But f(0) = sin0/0 is not defined

 

[snipez90]

Well I took it that the OP actually meant the sinc function, which just extends f(x) = sin(x)/x by continuity so that f(0) = 1.