limit x->0 (1-cos(x))/x^2 — step-by-step solution

1) Let's evaluate the limit of

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

when

x \to 0

The limit of

- \cos (x)

-1

when

x \to 0

The limit of

1 - \cos (x)

0

when

x \to 0

The limit of

x^{2}

0

⁺ when

x \to 0

For

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

, the limit takes the indeterminate form 0/0 when

x

→

0

2) We need a Taylor expansion for:

- \cos (x)

Apply the Taylor development

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

of cos with X=

x

around A=

0

(order N=2)

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

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

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

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

3) We redo the calculation using the Taylor expansion:

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

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

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

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

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

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

\frac{1}{2}

So the limit of

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

\frac{1}{2}

when

x \to 0

4) The approximate value of:

\frac{1}{2}

equals:

0.5

▷Answer: $\frac{1}{2}$

▷▷Numeric:

0.5