Daily Math · 2026-06-28

Question 1

The limit $\displaystyle\lim_{x 	o 0} \dfrac{1 - \cos x}{x^2}$ equals:

Accepted Answer

$\dfrac{1 - \cos x}{x^2} = \dfrac{2\sin^2(x/2)}{x^2} = \dfrac{1}{2} \cdot \!\left(\dfrac{\sin(x/2)}{x/2}ight)^2 	o \dfrac{1}{2}$.

Answer

$0$

Answer

$\dfrac{1}{2}$

Answer

$1$

Answer

$\infty$

Question 2

The set of units (multiplicatively invertible elements) of $\mathbb{Z}_8$ is:

Accepted Answer

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

Answer

$\{\hat{0}\}$

Answer

$\{\hat{1}, \hat{2}, \hat{3}\}$

Answer

$\{\hat{1}, \hat{3}, \hat{5}, \hat{7}\}$

Answer

$\mathbb{Z}_8 \setminus \{\hat{0}\}$

Question 3

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

Accepted Answer

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

Answer

$9$

Answer

$6$

Answer

$12$

Answer

$24$

Problems & worked solutions