Граница на функция

[Olivia]

1.
lim[tex]\frac{sin2x}{xcosx }[/tex]
x->0
2.
lim [tex]\frac{x^3 - 1}{ x^3 - x^2 - 3x +3}[/tex]
x->1
3.
lim [tex]\frac{2x+3}{x^2 - x + 1 }[/tex]
x-> - ?

[martin123456]

1
[tex]\lim_{x \rightarrow 0}\frac{sin2x}{xcosx }=\lim_{x \rightarrow 0}\frac{2\sin{x}\cos{x}}{x\cos{x}}=\frac{2\sin{x}}{x}=2[/tex]

[martin123456]

2
[tex]\lim_{x \rightarrow 1}\frac{x^3 - 1}{ x^3 - x^2 - 3x +3}=\lim_{x \rightarrow 1}\frac{(x-1)(x^2+x+1)}{x^2(x-1)-3(x-1)}=\lim_{x \rightarrow 1}\frac{(x-1)(x^2+x+1)}{(x-1)(x^2-3)}=\lim_{x \rightarrow 1}\frac{x^2+x+1}{x^2-3}=\frac{3}{-2}[/tex]

[martin123456]

3
[tex]\lim_{x \rightarrow -\infty}\frac{2x+3}{x^2 - x + 1 }=\lim_{x \rightarrow -\infty}\frac{\frac{2}{x^2}+\frac{3}{x^2}}{1-\frac{1}{x}+\frac{1}{x^2}}=\frac{0}{1}=0[/tex]