Question

Find the limit of the given problem in the picture:


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Answer 1

`~~\lim \limits_(x \to 0) \left((1- \cos mx )/(1- \cos nx)} \right)\\\\\\=\lim \limits_(x \to 0) \left[(2\sin^2 \left((mx)/(2) \right))/(2 \sin^2 \left((nx)/(2)\right)) \right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[1-\cos 2x =2 \sin^2 x\right]\\\\\\`

`=\lim \limits_(x \to 0) \left[ \frac{\sin^2 \left(\frac{mx}2 \right)}{\left(\frac{mx}2 \right)^2 } } * \left((mx)/(2) \right)^2* \frac{\left(\frac{nx}2 \right)^2} {\sin^2 \left(\frac{nx}2 \right)} * \left((2)/(nx) \right)^2\right]\\\\\\=\lim \limits_(x \to 0)\left[ \left( \frac{mx}2 \right)^2 \left( (2)/(nx) \right)^2\right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[\lim \limits_(x \to 0) (\sin x)/(x) = \lim \limits_(x \to 0) (x)/( \sin x) = 1\right] \\\\\\`

`=\lim \limits_(x \to 0) \left( (m^2 x^2)/(4) \cdot (4)/(n^2 x^2 ) \right)\\\\\\=\lim \limits_(x \to 0) \left( (m^2)/(n^2)\right)\\ \\\\=(m^2)/(n^2)`