Question

need help in limits

Answer 1

\[\lim_{n \rightarrow \infty} A^n(1/n)\]
\[\left[ \left[\begin{matrix}cosx & sinx \\ -sinx & cosx\end{matrix}\right] \right]=A\]

Answer 2

Is \(\left[ \left[\begin{matrix}cosx & sinx \\ -sinx & cosx\end{matrix}\right] \right]=A\) a determinant?

Answer 3

yes

Answer 4

If it is, then A=\(\cos^2(x)+\sin^2(x)\)=1

Answer 5

Can you pick up the rest?

Answer 6

sorry its not determinant its a matrix

Answer 7

So is the result supposed to be a matrix?

Answer 8

yes

Answer 9

Have you multiplied a matrix before?

Answer 10

yes i did it many times

Answer 11

and i did it for above too but only upto n=1,2,3,4
and by squaring A we get A replacing x with 2x and respectively for cubing and for n=4

Answer 12

So you found a trend!

Answer 13

If you can show that A^n (x) = A(nx), then the problem is solved!

Answer 14

In fact, if you can show that there is an upper bound for A^n, you can evaluate the limit as well, since the denominator is n->\(\infty\)

Answer 15

Hint:
Recall A is the transformation matrix for a rotation through angle x, and A^2 is applying the transformation 2 times, A^n is applying the transformation n times, hence your result of 2x, 3x, ...
Since the range of cos(x) and sin(x) are [-1,1], so the limit......