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

Main topic:

Limits

Related formulas:

7. See formulas

Time to solve it:

~ 0.04 s (SnapXam)