<s>[INST] computing a trignometric limit I am trying to show that $$\lim _{n\rightarrow \infty }\dfrac {1+\cos \dfrac {x} {n}+\cos \dfrac {2x} {n}+\ldots +\cos\dfrac {\left( n-1\right) x} {n}} {n } = \dfrac{\sin x}{x}$$
I have attempted a number of approaches such as trigonometric tricks and the most recent with substitution of $\cos \dfrac {x} {n}=\dfrac {e^{i\frac {x} {n}}+e^{-\frac {ix} {n}}} {2}$
Starting from the left hand side i ended up with an expression such as $$\lim _{n\rightarrow \infty }\frac {\frac {1-e^{ix}} {1-e^{\frac {ix} {n}}}+\frac {1-e^{-ix}} {1-e^{\frac {-ix} {n}}}} {2n}$$ I am unsure how i could possibly rearrange this so upon taking the limit i can convert to RHS.
Any help with this method or even an alternative proof strategy would be much appreciated.
 [/INST] Here is slightly long winded solution using trigonometric identities only.
$$\lim _{n\rightarrow \infty }\dfrac {1+\cos \dfrac {x} {n}+\cos \dfrac {2x} {n}+\ldots +\cos\dfrac {\left( n-1\right) x} {n}} {n } $$
using the trignometric identity $$\sum _{n=1}^{N}\cos n\theta =\dfrac {-1} {2}+\dfrac {\sin \left( N+\dfrac {1} {2}\right) \theta } {2\sin \dfrac {\theta } {2}}$$
our equation can be rewritten as 
$$\lim _{n\rightarrow \infty }\dfrac {1-\dfrac {1} {2}+\dfrac {\sin \left( n-1+\dfrac {1} {2}\right) \dfrac {x} {n}} {2\sin x / 2n}} {n}$$
$$\lim _{n\rightarrow \infty }\dfrac {1} {2n}\left( \dfrac {\sin\dfrac {x} {2n}+\sin \left( x-\dfrac {x} {2n}\right) } {\sin \dfrac {x} {2n}}\right) $$
$$\lim _{n\rightarrow \infty }\dfrac {1} {2n}\left( \dfrac {2\sin\dfrac {x} {2}\cos \left(\dfrac {\dfrac {x} {n}-x} {2}\right)} {\sin \dfrac {x} {2n}}\right) $$
The following restructure of the denominator was pointed out to me by Aryabhata. :-)
$$\lim _{n\rightarrow \infty }\dfrac {\sin \dfrac {x} {2}\cos \dfrac {x} {2}\left( \dfrac {1} {n}-1\right) } {\dfrac {x\sin \dfrac {x} {2n}} {2\dfrac {x} {2n}}}$$
Taking the limit we get
$$\dfrac {\sin \dfrac {x} {2}\cos -\dfrac {x} {2}} {\dfrac {x} {2}}$$
which is the same as 
$$\dfrac {\sin \dfrac {x} {2}\cos \dfrac {x} {2}} {\dfrac {x} {2}}$$
which gives RHS using the trigonometric identity once again.
$$\dfrac{\sin x}{x}$$
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