Question

Пожайлуста решите задания с пределами

Answer 1

$lim_{x to 2} frac{x^2-4}{x-2} = lim_{x to 2} frac{(x-2)(x+2)}{x-2} = lim_{x to 2} x+2==4$
$lim_{x to 1} frac{x^2-3x+2}{x^2+x-2} = lim_{x to 1} frac{(x-1)(x-2)}{(x-1)(x+2)} = lim_{x to 1} frac{x-2}{x+2}= frac{1-2}{1+2} = -frac{1}{3}$
$lim_{x to 2} frac{x^2-7x+10}{x^3-8} = lim_{x to 2} frac{(x-2)(x-5)}{(x-2)(x^2+2x+4)} = lim_{x to 2} frac{x-5}{x^2+2x+4} = \ frac{2-5}{2^2+2*2+4} = -frac{3}{12} =- frac{1}{4}$

$lim_{x to 0} frac{sin2x}{4x} = frac{1}{2} lim_{x to 0} frac{sin2x}{2x} = frac{1}{2}$
$lim_{x to 0} frac{sin12x}{tg4x} = lim_{x to 0} frac{ 12xfrac{sin12x}{12x} }{ 4xfrac{tg4x}{4x} } = frac{12}{4} =3$
$lim_{x to infty} (1+ frac{1}{2x} ) ^x= lim_{x to infty} ( (1+ frac{1}{2x} ) ^{2x})^{1/2}=e^{1/2}= sqrt{e}$
$lim_{x to 0} (1-2x)^{1/x}= lim_{x to 0}( (1+(-2x))^{-1/2x})^{-2}=e^{-2}= frac{1}{e^2}$