# Step-by-step Solution

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$\frac{1}{2}$$\,\,\left(\approx 0.5\right) ## Step-by-step explanation Problem to solve: \lim_{x\to\:0}\left(\frac{1-\cos\left(x\right)}{x^2}\right) Choose the solving method 1 If we directly evaluate the limit \lim_{x\to 0}\left(\frac{1-\cos\left(x\right)}{x^2}\right) as x tends to 0, we can see that it gives us an indeterminate form \frac{0}{0} 2 We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately \lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}{\frac{d}{dx}\left(x^2\right)}\right) 3 After deriving both the numerator and denominator, the limit results in \lim_{x\to0}\left(\frac{\sin\left(x\right)}{2x}\right) 4 If we directly evaluate the limit \lim_{x\to 0}\left(\frac{\sin\left(x\right)}{2x}\right) as x tends to 0, we can see that it gives us an indeterminate form \frac{0}{0} 5 We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately \lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{\frac{d}{dx}\left(2x\right)}\right) 6 After deriving both the numerator and denominator, the limit results in \lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right) 7 The limit of the product of a function and a constant is equal to the limit of the function, times the constant: \displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)} \frac{1}{2}\lim_{x\to0}\left(\cos\left(x\right)\right) 8 Evaluate the limit \lim_{x\to0}\left(\cos\left(x\right)\right) by replacing all occurrences of x by 0 \cos\left(0\right) 9 Simplifying, we get 1 10 Multiply \frac{1}{2} times 1 \frac{1}{2} ## Final Answer \frac{1}{2}$$\,\,\left(\approx 0.5\right)$
$\lim_{x\to\:0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$