Limit of sin(x)/x as x Approaches 0: Does it Equal 1?

[Miike012]

lim x → 0 sin(x)/x = 1

This doesn't mean that f '(0) = 1 if f(x) = sin(x)/x does it?

 

[Mark44]

lim x → 0 sin(x)/x = 1

This doesn't mean that f '(0) = 1 if f(x) = sin(x)/x does it?

No, it's just a limit that's not related to a derivative.

 

[Miike012]

But derivates and limits are similar correct?

 

[Mark44]

A derivative is a limit, but a limit is not necessarily a derivative.

 

[Dick]

lim x → 0 sin(x)/x = 1

This doesn't mean that f '(0) = 1 if f(x) = sin(x)/x does it?

That's a difference quotient for f(x)=sin(x). Not for f(x)=sin(x)/x.

 

[Harrisonized]

Definition of a derivative:

df/dx = lim_Δx→0 [f(x+Δx)-f(x)]/Δx

Changing the variables around, we find:

df/dy = lim_x→0 [f(y+x)-f(y)]/x

If lim_x→0 f(y+x)-f(y) = sin(x), then your relation holds. Not really sure what function would satisfy this though. Maybe you can tell me. :P

 

[HallsofIvy]

lim x → 0 sin(x)/x = 1

This doesn't mean that f '(0) = 1 if f(x) = sin(x)/x does it?

It certainly can be used to show that:

$$\frac{dsin}{dx}(0)= \lim_{x\to 0}\frac{sin(x)- sin(0)}{x}= \lim_{x\to 0} \frac{sin(x)}{x}$$.