Question

найти указанные пределы

Answer 1

1.$lim_{x to 2} frac{sqrt{x+2}-2}{x^{2}-4}= lim_{x to 2} frac{x-2}{(x^{2}-4)(sqrt{x+2}+2)}=$$lim_{x to 2} frac{1}{(x+2)(sqrt{x+2}+2)}= frac{1}{(2+2)(2+2)} = frac{1}{16}$
2.$lim_{x to0} frac{1-cos4x}{x^{2}}= lim_{x to0}frac{2sin^{2}2x}{x^{2}}=$$lim_{x to0}2*4*frac{sin2x*sin2x}{2x*2x}=8*1=8$
3.$lim_{x to infty} (frac{3x-4}{3x-2})^{6x+1}=lim_{x to infty} (1+frac{-2}{3x-2})^{6x+1}=$$lim_{x to infty} (1+frac{-2}{3x-2})^{ frac{3x-2}{-2}frac{-2}{3x-2} 6x+1}=$$e^{lim_{x to infty}frac{-2}{3x-2}6x+1}=e^{ frac{-12}{3} }=e^{-4}= frac{1}{e^4}$