What is the limit as x approaches+0 of ((cos(mx)-cos(nx)))/(x^2) ?

The limit as x approaches+0 of ((cos(mx)-cos(nx)))/(x^2) is (-m^2+n^2)/2

limit as x approaches+0 of ((cos(mx)-cos(nx)))/(x^2)

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

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

Solution steps

x \to 0 lim (\frac{cos ( m x ) - cos ( n x )}{x ^{2}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{- m sin ( m x ) + n sin ( n x )}{2 x})

Apply L'Hopital's Rule

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

Plug in the value

x = 0

= \frac{- m ^{2} cos ( m \cdot 0 ) + n ^{2} cos ( n \cdot 0 )}{2}

- m^{2} cos (m \cdot 0) + n^{2} cos (n \cdot 0) = - m^{2} + n^{2}

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

x \to 0 lim (4 x - 8)

x \to \frac{1}{2} + lim (x^{2} tan (π x))

x \to - 1 lim (\frac{x}{∣ x + 1 ∣})

x \to - 2 - lim (h (x))

x \to 2 + lim (- \frac{1}{( x - 2 ) ^{2}})

What is the limit as x approaches+0 of ((cos(mx)-cos(nx)))/(x^2) ?
The limit as x approaches+0 of ((cos(mx)-cos(nx)))/(x^2) is (-m^2+n^2)/2