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

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

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Answer

$\frac{1}{2}$

Step by step solution

Problem

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

As the limit results in indeterminate form, we can apply L'Hôpital's rule

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(-\cos\left(x\right)\right)+\frac{d}{dx}\left(1\right)}{2x}\right)$
4

The derivative of the constant function 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 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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to0}\left(\frac{-1\left(-1\right)\sin\left(x\right)}{2x}\right)$
7

Multiply $-1$ times $-1$

$\lim_{x\to0}\left(\frac{1\sin\left(x\right)}{2x}\right)$
8

Any expression multiplied by $1$ is equal to itself

$\lim_{x\to0}\left(\frac{\sin\left(x\right)}{2x}\right)$
9

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{\frac{d}{dx}\left(2x\right)}\right)$
10

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{2\frac{d}{dx}\left(x\right)}\right)$
11

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{2}\right)$
12

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$
13

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{2}\lim_{x\to0}\left(\cos\left(x\right)\right)$
14

Evaluating the limit when $x$ tends to $0$

$\cos\left(0\right)\cdot \frac{1}{2}$
15

Simplifying

$\frac{1}{2}$

Answer

$\frac{1}{2}$

Problem Analysis

Main topic:

Limits by L'Hôpital's rule

Time to solve it:

0.22 seconds

Views:

199