The limit lim_x → 0 (1 - cos x)/(x^2) equals:

(1 - cos x)/(x^2) = (2sin^2(x/2))/(x^2) = (1)/(2) · ((sin(x/2))/(x/2))^2 → (1)/(2).

Daily Math Problems — June 28, 2026: Calculus, Algebra, Combinatorics

Problems & worked solutions

RO M1

The limit $\lim_{x \to 0} \frac{1 - \cos x}{x^{2}}$ equals:

Show answer & worked solution

$\frac{1 - \cos x}{x^{2}} = \frac{2 \sin^{2} (x / 2)}{x^{2}} = \frac{1}{2} \cdot ⁣ {(\frac{\sin (x / 2)}{x / 2})}^{2} \to \frac{1}{2}$ .

RO M1

The set of units (multiplicatively invertible elements) of $Z_{8}$ is:

Show answer & worked solution

The units are the residues coprime to $8$ : ${1, 3, 5, 7}$ , four elements.

International

The number of permutations of $(1, 2, 3, 4)$ in which no element is in its original position is:

Show answer & worked solution

$! 4 = 4! (1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}) = 24 \cdot \frac{9}{24} = 9$ .

Problems & worked solutions

RO M1

The limit $\lim_{x \to 0} \frac{1 - \cos x}{x^{2}}$ equals:

Show answer & worked solution

$\frac{1 - \cos x}{x^{2}} = \frac{2 \sin^{2} (x / 2)}{x^{2}} = \frac{1}{2} \cdot ⁣ {(\frac{\sin (x / 2)}{x / 2})}^{2} \to \frac{1}{2}$ .

RO M1

The set of units (multiplicatively invertible elements) of $Z_{8}$ is:

Show answer & worked solution

The units are the residues coprime to $8$ : ${1, 3, 5, 7}$ , four elements.

International

The number of permutations of $(1, 2, 3, 4)$ in which no element is in its original position is:

Show answer & worked solution

$! 4 = 4! (1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}) = 24 \cdot \frac{9}{24} = 9$ .