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

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

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

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

\frac{a ^{2}}{2}

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

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

= \frac{a ^{2}}{2}

\frac{d}{d x} (sin^{3} (\frac{x}{3}))

\int \frac{x}{x + 2} d x

n = 1 \sum \infty \frac{n !}{2 n !}

\frac{d}{d x} (\frac{2}{x ^{5}})

f (x) 2 z (9 z^{2} - 3 z + 1)^{4}

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