Question

Найти предел и производные. Помогите пожалуйста)

Answer 1

$\ lim_{xrightarrowpi}{1-cos{3x}over ctgx}=lim_{xrightarrowpi}{sin x(1-cos{3x})over cos x}={sin pi(1-cos{3pi})over cos pi}={0*(1+1)over-1}=0$

$\ lim_{xrightarrow infty}{2x^2-5x+8over3x^2+6x-15}=lim_{xrightarrow infty}{2x^2over3x^2}={2over3}$


$\ lim_{xrightarrow -2}{2x^2+7x+6 over x^2+x-2}=left | {0over0} right |=*\\2x^2+7x+6=0\D=49-48=1\x_1={-7+1over4}={-3over2}=-1,5\x_2={-7-1over4}=-2\x^2+x-2=0\x_1=-2, x_2=1\\*=lim_{xrightarrow -2}{(x+2)(2x+3)over(x+2)(x-1)}=lim_{xrightarrow -2}{(2x+3)over(x-1)}={-4+3over-3}={1over 3}$

$\lim_{xrightarrow 1}(3-2x^2)^{1over2(1-x)}=*\ left [lim_{xrightarrow a}(1+u)^{1over u}=e\ right ]\ u=2-2x^2\ {1over u}={1over{2-2x^2}}\ \*=lim_{xrightarrow 1}(1+(2-2x^2))^{{1over2(1-x)}*{2(1-x^2)over2(1-x^2)}}=lim_{xrightarrow 1}(1+(2-2x^2))^{{1+xover2(1-x^2)}}=lim_{xrightarrow 1}e^{1+x}=e^2$