Step-by-step Solution

Evaluate the limit of $\frac{1-\cos\left(x\right)}{x^2}$ as $x$ approaches 0

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Final Answer

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

Main topic:

Limits

Time to solve it:

~ 0.22 s (SnapXam)