Limits

[monomocoso]

How to show

lim n-> infinity of sin (pi/4n) = 0 and
lim n-> infinity of cos (pi/4n) = 1
?

 

[pappus]

How to show

lim n-> infinity of sin (pi/4n) = 0 and
lim n-> infinity of cos (pi/4n) = 1
?

1. Let a denotes the argument of the Sine function. Then \(\displaystyle a \in (0,\frac \pi4)\)
Show that the Sine function is monotonely(?) increasing over the interval of a.

2. Now you can use \(\displaystyle \lim_\csub{n \to \infty}\sin\left(\frac \pi{4n} \right) = \sin\left(\lim_{n \to \infty}\left(\frac \pi{4n}\right) \right) = \sin(0) = 0\)

3. The second problem should be done similarly.