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

1) Compute the limit of

\frac{x^{2}}{1 - \cos (x)}

x \to 0

The limit of

x^{2}

0

⁺ when

x \to 0

ⓘ The function cos is continuous, its limit is its value at the limit of its argument

The limit of

- \cos (x)

- \cos (0)

⁺ when

x \to 0

⇥1.1) Let's evaluate:

- \cos (0)

⇥ⓘ The cosine is the x-coordinate (horizontal axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, this is a remarkable value:

- \cos (0)

-1

⇥

-1

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

1 - \cos (x)

is (

1 - 1

)⁺ when

x \to 0

⇥1.2) Let's calculate:

1 - 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

For

\frac{x^{2}}{1 - \cos (x)}

, we encounter the indeterminate form 0/0 as

x

tends to

0

2) We need a Taylor expansion for:

- \cos (x)

Using Taylor's formula:

\cos (A) - \sin (A) \cdot (X - A) - \cos (A) \cdot \frac{{(X - A)}^{2}}{2}

for the function cos, we compute the Taylor expansion for X=

x

at the point A=

0

of order N=2

\cos (0) - \sin (0) \cdot (x + 0) - \cos (0) \cdot \frac{{(x + 0)}^{2}}{2}

ⓘ The cosine is the x-coordinate (horizontal axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, this is a remarkable value:

\cos (0)

1

- \cos (0)

-1

■The sine is the y-coordinate (vertical axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, this is a remarkable value:

\sin (0)

0

■Adding 0 changes nothing:

x + 0

x

■Multiplying a fraction by a number multiplies its numerator:

-1 \cdot \frac{x^{2}}{2}

\frac{-1 \cdot x^{2}}{2}

- \frac{x^{2}}{2}

1 + 0 \cdot x - \frac{x^{2}}{2}

ⓘ A product with a zero factor is zero:

0 \cdot x

0

1 + 0 - \frac{x^{2}}{2}

ⓘ Adding 0 changes nothing:

0 - \frac{x^{2}}{2}

- \frac{x^{2}}{2}

1 - \frac{x^{2}}{2}

3) We redo the calculation using the Taylor expansion:

- 1 + \frac{x^{2}}{2}

\frac{x^{2}}{1 + (- 1 + \frac{x^{2}}{2})}

ⓘ The numerator and the denominator are computed before the division:

1 - 1

0

■To add fractions, put them over the same denominator, then add the numerators:

0 + \frac{x^{2}}{2}

\frac{0 \cdot 2 + x^{2}}{2}

\frac{x^{2}}{\frac{0 \cdot 2 + x^{2}}{2}}

ⓘ A product with a zero factor is zero:

0 \cdot 2

0

\frac{x^{2}}{\frac{0 + x^{2}}{2}}

ⓘ Adding 0 changes nothing:

0 + x^{2}

x^{2}

\frac{x^{2}}{\frac{x^{2}}{2}}

ⓘ Dividing is multiplying by the reciprocal:

\frac{x^{2}}{\frac{x^{2}}{2}}

x^{2} \cdot \frac{2}{x^{2}}

x^{2} \cdot \frac{2}{x^{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

x^{2} \cdot \frac{2}{x^{2}}

\frac{x^{2} \cdot 2}{x^{2}}

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x^{2}

\frac{x^{2} \cdot 2}{x^{2}}

2

2

So the limit of

\frac{x^{2}}{1 - \cos (x)}

2

when

x \to 0

▶Limit:

2