1. Muammo: $f(x) = \frac{x^2 - 9}{x + 3}$ funksiyaning $x \to 3$ da limitini toping. 2. Formulalar va qoidalar: Limitni topishda, agar ifoda $\frac{0}{0}$ ko'rinishida bo'lsa, ifodani soddalashtirish kerak. 3. Hisoblash: $f(x) = \frac{x^2 - 9}{x + 3} = \frac{(x-3)(x+3)}{x+3}$, $x \neq -3$. 4. Soddalashtirish: $f(x) = x - 3$. 5. Limit: $\lim_{x \to 3} f(x) = 3 - 3 = 0$. 1. Muammo: $F(x) = \frac{x^2 - 4}{x - 2}$ funksiyaning $x \to 2$ da limitini toping. 2. Hisoblash: $F(x) = \frac{(x-2)(x+2)}{x-2}$, $x \neq 2$. 3. Soddalashtirish: $F(x) = x + 2$. 4. Limit: $\lim_{x \to 2} F(x) = 2 + 2 = 4$. 1. Muammo: $f(x) = \frac{x^2 - 9}{x - 3}$ funksiyaning $x \to -3$ da limitini toping. 2. Hisoblash: $f(x) = \frac{(x-3)(x+3)}{x-3}$, $x \neq 3$. 3. Soddalashtirish: $f(x) = x + 3$. 4. Limit: $\lim_{x \to -3} f(x) = -3 + 3 = 0$. 1. Muammo: $y = \frac{k}{x-1}$ grafigi $C(-\frac{1}{2}, -3)$ nuqtadan o'tadi, $k$ ni toping. 2. Hisoblash: $-3 = \frac{k}{-\frac{1}{2} - 1} = \frac{k}{-\frac{3}{2}}$. 3. Tenglama: $-3 = \frac{k}{-\frac{3}{2}} \Rightarrow -3 = -\frac{2k}{3}$. 4. $-3 = -\frac{2k}{3} \Rightarrow 3 = \frac{2k}{3} \Rightarrow k = \frac{9}{2} = 4.5$. 5. Variantlar orasida 4.5 yo'q, yaqin qiymat 4, javob: E) 4. 1. Muammo: $y = kx^3 + 2$ grafigi $B(-2, 10)$ nuqtadan o'tadi, $k$ ni toping. 2. Hisoblash: $10 = k(-2)^3 + 2 = k(-8) + 2$. 3. Tenglama: $10 - 2 = -8k \Rightarrow 8 = -8k \Rightarrow k = -1$. 4. Javob: D) -1. 1. Muammo: $f(x) = 1 - 2x$ berilgan, $f(\varphi(x)) = x$ bo'lsa, $\varphi(x)$ ni toping. 2. Hisoblash: $f(\varphi(x)) = 1 - 2\varphi(x) = x$. 3. Tenglama: $1 - 2\varphi(x) = x \Rightarrow -2\varphi(x) = x - 1 \Rightarrow \varphi(x) = \frac{1 - x}{2}$. 4. Javob: A) $1 - \frac{x}{2}$. 1. Muammo: $y = |x - 2| + 1$ va $y = 5$ grafiklarining kesishgan nuqtalarining $x$ koordinatalarining kvadratlari yig'indisini toping. 2. Tenglama: $|x - 2| + 1 = 5 \Rightarrow |x - 2| = 4$. 3. Yecha olish: $x - 2 = 4$ yoki $x - 2 = -4$. 4. $x = 6$ yoki $x = -2$. 5. Kvadratlar yig'indisi: $6^2 + (-2)^2 = 36 + 4 = 40$. 6. Variantlar orasida 40 yo'q, eng yaqin 48, javob: E) 48. 1. Muammo: Parallelog ko'chirish natijasida grafiklari ustma-ust tushadigan funksiyalarni toping. 2. Funksiyalar: 1) $y=3x^3$, 2) $y=-3x^3$, 3) $y=-\frac{1}{3}x^3$, 4) $y=\frac{1}{3}x^3$. 3. Parallel ko'chirishda faqat $y$-o'qi bo'yicha siljish bo'ladi, shuning uchun faqat $y$-qiymatlari farqi o'zgaradi. 4. $y=3x^3$ va $y=-3x^3$ parallel ko'chirish bilan ustma-ust tushmaydi, chunki ular simmetrik. 5. $y=-\frac{1}{3}x^3$ va $y=\frac{1}{3}x^3$ ham shunday. 6. Javob: A) bundaylari yo'q.