Can we prove the limit of sin(x)/x is 1 using the ε-δ definition?

[evagelos]

possible proof??

Can we prove that the limit of :

lim_{n\to 0}\frac{sinx}{x} =1

By using the ε-δ definition??

 

[Feldoh]

http://en.wikipedia.org/wiki/Squeeze_theorem#Proof

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[rochfor1]

I don't believe the Squeeze Theorem works too well for this one, because \lim_{ x \to 0 } \pm \frac{1}{x} does not exist. The standard proof of this fact would cite either a geometric argument, Taylor series, or L'Hopital's theorem.

 

[dacruick]

i dunno, take the derivative and set x equal to 0 and you get 1. that's proof enough for a physics forum

 

[breedencm]

d/dx sin(x) @ x = 0, is defined as: lim x->0 [sin(x) - sin(0)]/x = lim x->0 sin(x)/x.

Basically, you can't assume that this limit is 1 to prove that this limit is 1, unless ofcourse you can prove that d/dx sin(x) = cos(x) and consider this as a special case (however, you will come to find that hidden in this proof contains the question being asked).