If-lim-x-0-cos-m-mx-cos-n-nx-m-2-n-2-mn-x-2-1-find-m-2-n-2-4-mn-

Question Number 168576 by cortano1 last updated on 13/Apr/22

I f \lim_{x \to 0} \frac{\cos^{m} (m x) - \cos^{n} (n x)}{(m^{2} + n^{2} + m n) x^{2}} = 1

f i n d \frac{m^{2} + n^{2} - 4}{m n} .

Commented by blackmamba last updated on 17/Apr/22

m^{2} + n^{2} + m n = \lim_{x \to 0} \frac{\cos^{m} (m x) - \cos^{n} (n x)}{x^{2}}

m^{2} + n^{2} + m n = \frac{n^{3} - m^{3}}{2}

\frac{n^{3} - m^{3}}{(n - m)} = \frac{n^{3} - m^{3}}{2}

(n^{3} - m^{3}) (\frac{1}{n - m} - \frac{1}{2}) = 0

n - m = 2 \Rightarrow n^{2} - 2 m n + m^{2} = 4

\Rightarrow m^{2} + n^{2} - 4 = 2 m n

\Rightarrow \frac{m^{2} + n^{2} - 4}{m n} = 2