Daily Math · 2026-06-14

Question 1

Find $\displaystyle\lim_{x	o\frac{\sqrt{3}}{2}}(\arcsin x+\arccos x)$.

Accepted Answer

$\bullet$ Use the complementary-angle identity, valid for every $x\in[-1,1]$:

$$\arcsin x+\arccos x=	frac{\pi}{2}$$

$\bullet$ The expression is a constant, so the limit is just that constant:

$$\lim_{x	o\sqrt{3}/2}(\arcsin x+\arccos x)=	frac{\pi}{2}$$

Answer

$\dfrac{\pi}{3}$

Answer

$\dfrac{\pi}{2}$

Answer

$\pi$

Answer

$\dfrac{2\pi}{3}$

Answer

$0$

Answer

$1$

Question 2

A field is a commutative ring with unity in which:

Accepted Answer

Field: commutative ring with unity such that every non-zero element has a multiplicative inverse.

Answer

there are no zero divisors

Answer

every element has an inverse

Answer

every non-zero element has a multiplicative inverse

Answer

$(R, +) \cong (R, \cdot)$

Question 3

The image of $f: \mathbb{R} 	o \mathbb{R}$, $f(x) = -x^2 + 4x - 1$, is:

Accepted Answer

$\Delta = 16 - 4 = 12$, $a = -1$, so $f_{\max} = -\dfrac{\Delta}{4a} = -\dfrac{12}{-4} = 3$. The image is $(-\infty, 3]$.

Answer

$\mathbb{R}$

Answer

$[3, +\infty)$

Answer

$(-\infty, 3]$

Answer

$(-\infty, -1]$

Problems & worked solutions