Дано: Вычислить пределы:
1) $\lim_{x \to 3} \frac{x^2 - 5x + 6}{x^2 - 8x + 15}$
2) $\lim_{x \to \infty} \frac{3x^2 - 4x}{4 + x^2}$
3) $\lim_{x \to 2} \frac{\sqrt{2x} - 2}{x^2 - 4x + 4}$
4) $\lim_{x \to 0} \frac{2x \sin x}{1 - \cos^2 2x}$
5) $\lim_{x \to \infty} (\frac{4x + 1}{4x - 1})^{x - 4}$

Решение:
1) $\lim_{x \to 3} \frac{x^2 - 5x + 6}{x^2 - 8x + 15} = \lim_{x \to 3} \frac{(x - 2)(x - 3)}{(x - 3)(x - 5)} = \lim_{x \to 3} \frac{x - 2}{x - 5} = \frac{3 - 2}{3 - 5} = \frac{1}{-2} = -\frac{1}{2}$

2) $\lim_{x \to \infty} \frac{3x^2 - 4x}{4 + x^2} = \lim_{x \to \infty} \frac{x^2(3 - \frac{4}{x})}{x^2(\frac{4}{x^2} + 1)} = \lim_{x \to \infty} \frac{3 - \frac{4}{x}}{\frac{4}{x^2} + 1} = \frac{3 - 0}{0 + 1} = 3$

3) $\lim_{x \to 2} \frac{\sqrt{2x} - 2}{x^2 - 4x + 4} = \lim_{x \to 2} \frac{\sqrt{2x} - 2}{(x - 2)^2} = \lim_{x \to 2} \frac{(\sqrt{2x} - 2)(\sqrt{2x} + 2)}{(x - 2)^2(\sqrt{2x} + 2)} = \lim_{x \to 2} \frac{2x - 4}{(x - 2)^2(\sqrt{2x} + 2)} = \lim_{x \to 2} \frac{2(x - 2)}{(x - 2)^2(\sqrt{2x} + 2)} = \lim_{x \to 2} \frac{2}{(x - 2)(\sqrt{2x} + 2)}$.
При $x \to 2$, $(x - 2) \to 0$, а $(\sqrt{2x} + 2) \to 4$. Следовательно, предел равен $\infty$.

4) $\lim_{x \to 0} \frac{2x \sin x}{1 - \cos^2 2x} = \lim_{x \to 0} \frac{2x \sin x}{\sin^2 2x} = \lim_{x \to 0} \frac{2x \sin x}{(2 \sin x \cos x)^2} = \lim_{x \to 0} \frac{2x \sin x}{4 \sin^2 x \cos^2 x} = \lim_{x \to 0} \frac{x}{2 \sin x \cos^2 x} = \lim_{x \to 0} \frac{x}{\sin x} \cdot \lim_{x \to 0} \frac{1}{2 \cos^2 x} = 1 \cdot \frac{1}{2 \cdot 1^2} = \frac{1}{2}$

5) $\lim_{x \to \infty} (\frac{4x + 1}{4x - 1})^{x - 4} = \lim_{x \to \infty} (\frac{4x - 1 + 2}{4x - 1})^{x - 4} = \lim_{x \to \infty} (1 + \frac{2}{4x - 1})^{x - 4} = \lim_{x \to \infty} (1 + \frac{2}{4x - 1})^{\frac{4x - 1}{2} \cdot \frac{2}{4x - 1} \cdot (x - 4)} = e^{\lim_{x \to \infty} \frac{2(x - 4)}{4x - 1}} = e^{\lim_{x \to \infty} \frac{2x - 8}{4x - 1}} = e^{\lim_{x \to \infty} \frac{2 - \frac{8}{x}}{4 - \frac{1}{x}}} = e^{\frac{2}{4}} = e^{\frac{1}{2}} = \sqrt{e}$

Ответ:
1) $-\frac{1}{2}$
2) $3$
3) $\infty$
4) $\frac{1}{2}$
5) $\sqrt{e}$