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

Step-by-step Solution

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Final Answer

$\frac{1}{2}$
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Step-by-step Solution

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We could not solve this problem by using the method: Limits by Rationalizing

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'H么pital'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'H么pital'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

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

$\frac{\cos\left(0\right)}{2}$
8

The cosine of $0$ equals $1$

$\frac{1}{2}$
9

Divide $1$ by $2$

$\frac{1}{2}$

Final Answer

$\frac{1}{2}$

Exact 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'H么pital's ruleLimits by Factoring

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Function Plot

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

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7
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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

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