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Step-by-step Solution

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

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Answer

$\frac{1}{2}$

Step-by-step explanation

Problem to solve:

$\lim_{x\to0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$
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As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}{\frac{d}{dx}\left(x^2\right)}\right)$
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The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}{2x}\right)$

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Answer

$\frac{1}{2}$
$\lim_{x\to0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$

Main topic:

Limits by L'Hôpital's rule

Used formulas:

5. See formulas

Time to solve it:

~ 0.69 seconds

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