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

Step-by-step Solution

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Solution

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Final Answer

$\frac{1}{2}$
Got another answer? Verify it here!

Step-by-step Solution

Problem to solve:

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

Specify the solving method

Plug in the value $0$ into the limit

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

The cosine of $0$ equals $1$

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

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

$\frac{1}{2}$

Numeric Answer

$0.5$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Limits by direct substitutionLimits by L'Hôpital's ruleLimits by factoringLimits by rationalizing
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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
D□x
∫
∫◻
|◻|
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

How to improve your answer:

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

Main topic:

Limits

Used formulas:

7. See formulas

Time to solve it:

~ 0.06 s

Related topics:

LimitsSum Rule of DifferentiationConstant Rule for DifferentiationBasic Differentiation RulesPower Rule for DerivativesLimits by L'Hôpital's rule

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