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Step-by-step Solution

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

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Final Answer

$\frac{1}{2}$$\,\,\left(\approx 0.5\right)$
Got a different answer? Try our new Answer Assistant!

Step-by-step Solution

Problem to solve:

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

Choose the solving method

Plug in the value $0$ into the limit

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

The cosine of $0$ equals $1$

$\frac{1-1\cdot 1}{0^2}$

Multiply $-1$ times $1$

$\frac{1-1}{0^2}$

Subtract the values $1$ and $-1$

$\frac{0}{0^2}$

Calculate the power $0^2$

$\frac{0}{0}$
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

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

Find the derivative of the numerator

$\frac{d}{dx}\left(1-\cos\left(x\right)\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

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

The derivative of the constant function ($1$) is equal to zero

$\frac{d}{dx}\left(-\cos\left(x\right)\right)$

The derivative of a function multiplied by a constant ($-1$) is equal to the constant times the derivative of the function

$-\frac{d}{dx}\left(\cos\left(x\right)\right)$

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\sin\left(x\right)$

Find the derivative of the denominator

$\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$
3

After deriving both the numerator and denominator, the limit results in

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

Plug in the value $0$ into the limit

$\frac{\sin\left(0\right)}{2\cdot 0}$

The sine of $0$ equals $0$

$\frac{0}{2\cdot 0}$

Multiply $2$ times $0$

$\frac{0}{0}$
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

$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)}{\frac{d}{dx}\left(2x\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(\sin\left(x\right)\right)$

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

$\cos\left(x\right)$

Find the derivative of the denominator

$\frac{d}{dx}\left(2x\right)$

The derivative of the linear function times a constant, is equal to the constant

$2$
6

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$
7

Evaluate the limit $\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$ by replacing all occurrences of $x$ by $0$

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

The cosine of $0$ equals $1$

$\frac{1}{2}$

Divide $1$ by $2$

$\frac{1}{2}$
8

Simplifying, we get

$\frac{1}{2}$

Final Answer

$\frac{1}{2}$$\,\,\left(\approx 0.5\right)$
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Got another answer? Verify it!

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1
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4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Tips on how to improve your answer:

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

Related Formulas:

7. See formulas

Time to solve it:

~ 0.06 s