What is the limit as x approaches 0 of (1-cos(kx))/(x^2) ?

The limit as x approaches 0 of (1-cos(kx))/(x^2) is (k^2)/2

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limit as x approaches 0 of (1-cos(kx))/(x^2)

x \to 0 lim (\frac{1 - cos ( k x )}{x ^{2}})

\frac{k ^{2}}{2}

Solution steps

x \to 0 lim (\frac{1 - cos ( k x )}{x ^{2}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{k sin ( k x )}{2 x})

= x \to 0 lim (\frac{k sin ( k x )}{2 x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{k ^{2} cos ( k x )}{2})

= x \to 0 lim (\frac{k ^{2} cos ( k x )}{2})

Plug in the value

x = 0

= \frac{k ^{2} cos ( k \cdot 0 )}{2}

k^{2} cos (k \cdot 0) = k^{2}

= \frac{k ^{2}}{2}

x \to - 3 lim (\frac{1}{9 - x ^{2}})

x \to 3 lim (x^{2} - 2 x + 3)

t \to 3 lim (\frac{1}{t - 3})

x \to 0 lim (x + x^{2})

x \to - 1 lim ((x + 3 x^{5})^{4})

What is the limit as x approaches 0 of (1-cos(kx))/(x^2) ?
The limit as x approaches 0 of (1-cos(kx))/(x^2) is (k^2)/2