Solve the following limit problems: 5. lim x+2 / x-5 as x approaches -2 6. lim 5+2x^2+x / x^2+3x-2 as x approaches -2 7. lim (4+x/2) as x approaches -2 8. lim 4-a / a+1 as a approaches 4 9. lim a^2+3a-3 / 6-a as a approaches 3

Решение:

• 5. \[ \lim_{x \to -2} \frac{x+2}{x-5} = \frac{-2+2}{-2-5} = \frac{0}{-7} = 0 \]
• 6. \[ \lim_{x \to -2} \frac{5+2x^2+x}{x^2+3x-2} = \frac{5+2(-2)^2+(-2)}{(-2)^2+3(-2)-2} = \frac{5+2(4)-2}{4-6-2} = \frac{5+8-2}{4-8} = \frac{11}{-4} = -2.75 \]
• 7. \[ \lim_{x \to -2} \left(4+\frac{x}{2}\right) = 4+\frac{-2}{2} = 4+(-1) = 3 \]
• 8. \[ \lim_{a \to 4} \frac{4-a}{a+1} = \frac{4-4}{4+1} = \frac{0}{5} = 0 \]
• 9. \[ \lim_{a \to 3} \frac{a^2+3a-3}{6-a} = \frac{3^2+3(3)-3}{6-3} = \frac{9+9-3}{3} = \frac{15}{3} = 5 \]

Ответ:

• 5. 0
• 6. -2.75
• 7. 3
• 8. 0
• 9. 5