Question

Вычислить пределы функций

Answer 1

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

2)
$\lim_{x \to 3} \frac{3x^2-10x+3}{x^2-2x-3} = \lim_{x \to 3} \frac{(3x-1)(x-3)}{(x+1)(x-3)} =\lim_{x \to 3} \frac{3x-1}{x+1} = \frac{8}{4}=2$

3)
$\lim_{x \to 3} \frac{x-3}{ \sqrt{4x-3} -3} = \lim_{x \to 3} \frac{(x-3)(\sqrt{4x-3} +3)}{ (\sqrt{4x-3} -3)(\sqrt{4x-3} +3)} =\\\lim_{x \to 3} \frac{(x-3)(\sqrt{4x-3} +3)}{4x-12} =\lim_{x \to 3} \frac{(x-3)(\sqrt{4x-3} +3)}{4(x-3)} =\\\lim_{x \to 3} \frac{\sqrt{4x-3} +3}{4} = \frac{3+3}{4} =1.5$