# Step-by-step Solution

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$\frac{1}{2}$$\,\,\left(\approx 0.5\right) ## Step-by-step explanation Problem to solve: \lim_{x\to 0}\left(\frac{1-\cos\left(x\right)}{x^2}\right) Choose the resolution method 1 If we try to evaluate the limit directly, it results in indeterminate form. Then we need to 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) 2 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) 3 The derivative of a sum of two functions is the sum of the derivatives of each function \lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-\cos\left(x\right)\right)}{2x}\right) 4 The derivative of the constant function (1) is equal to zero \lim_{x\to0}\left(\frac{\frac{d}{dx}\left(-\cos\left(x\right)\right)}{2x}\right) 5 The derivative of a function multiplied by a constant (-1) is equal to the constant times the derivative of the function \lim_{x\to0}\left(\frac{-\frac{d}{dx}\left(\cos\left(x\right)\right)}{2x}\right) 6 The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x) \lim_{x\to0}\left(\frac{\sin\left(x\right)}{2x}\right) 7 If we try to evaluate the limit directly, it results in indeterminate form. Then we need to apply L'Hôpital's rule \lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{\frac{d}{dx}\left(2x\right)}\right) 8 The derivative of the linear function times a constant, is equal to the constant \lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{2}\right) 9 The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)} \lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right) 10 The limit of the product of a function and a constant is equal to the limit of the function, times the constant: \displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)} \frac{1}{2}\lim_{x\to0}\left(\cos\left(x\right)\right) 11 Evaluate the limit by replacing all occurrences of x by 0 \frac{1}{2}\cos\left(0\right) 12 Simplifying \frac{1}{2}\cdot 1 13 Multiply \frac{1}{2} times 1 \frac{1}{2} ## Final Answer \frac{1}{2}$$\,\,\left(\approx 0.5\right)$

### Problem Analysis

$\lim_{x\to 0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$

Limits

~ 0.07 seconds