Задача 43203 ...

vk72207933

2019-12-29 18:28:21

Вычислить предел

[block]lim_(x → -3) (sqrt(1-5x)-sqrt(7-3x))/(x^2-2x-15)[/block]

sova

2019-12-29 20:07:21

[m]\lim_{x \to-3}\frac{\sqrt{1-5x}-\sqrt{7-3x}}{x^2-2x-15}=\frac{0}{0}[/m]

Умножаем и числитель и знаменатель на
(sqrt(1-5x)+sqrt(7-3x))

[m]=\lim_{x \to-3}\frac{(\sqrt{1-5x}-\sqrt{7-3x})(\sqrt{1-5x}+\sqrt{7-3x})}{(x^2-2x-15)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

Применяем формулу разности квадратов a^2-b^2=(a-b)*(a+b)


[m]\lim_{x \to-3}\frac{(\sqrt{1-5x})^2-(\sqrt{7-3x})^2}{(x^2-2x-15)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

[m]=\lim_{x \to-3}\frac{(1-5x)-(7-3x)}{(x^2-2x-15)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

[m]=\lim_{x \to-3}\frac{1-5x-7+3x}{(x+3)(x-5)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

[m]=\lim_{x \to-3}\frac{-2x-6}{(x+3)(x-5)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

[m]=\lim_{x \to-3}\frac{-2(x+3)}{(x+3)(x-5)(\sqrt{1-5x}+\sqrt{7-3x})}=[/m]

[m]=\lim_{x \to-3}\frac{-2}{(x-5)(\sqrt{1-5x}+\sqrt{7-3x})}=\frac{-2}{(-3-5)(\sqrt{1-5\cdot(-3)}+\sqrt{7-3\cdot(-3)})}=\frac{1}{32}[/m]