Przykład 1.
[tex]\lim_{x \to7} \frac{2x^2-11x-21}{x^2-9x+14}=\lim_{x \to7} \frac{2x^2-14x+3x-21}{x^2-7x-2x+14}=\lim_{x \to7} \frac{2x(x-7)+3(x-7)}{x(x-7)-2(x-7)}= \lim_{x \to7} \frac{(x-7)(2x+3)}{(x-7)(x-2)}=\lim_{x \to7} \frac{2x+3}{x-2}=\frac{2*7+3}{7-2}=\frac{17}{5}[/tex]
Przykład 2.
[tex]\lim_{x \to 1} \left(\frac{3}{1-x^3}+\frac{1}{x-1}\right)= \lim_{x \to 1} \left(\frac{3}{(1-x)(1+x+x^2)}+\frac{1}{x-1}\right)=\\=\lim_{x \to 1} \left(\frac{-3}{(x-1)(1+x+x^2)}+\frac{1+x+x^2}{(x-1)(1+x+x^2)}\right)=\lim_{x \to 1} \frac{-3+1+x+x^2}{(x-1)(1+x+x^2)}=\\=\lim_{x \to 1} \frac{x^2+x-2}{(x-1)(1+x+x^2)}=\lim_{x \to 1} \frac{x^2-x+2x-2}{(x-1)(1+x+x^2)}=\lim_{x \to 1} \frac{x(x-1)+2(x-1)}{(x-1)(1+x+x^2)}=[/tex]
[tex]=\lim_{x \to 1} \frac{(x-1)(x+2)}{(x-1)(1+x+x^2)}=\lim_{x \to 1} \frac{x+2}{1+x+x^2}=\frac{1+2}{1+1+1^2}=\frac{3}{3}=1[/tex]
Przykład 3.
[tex]\lim_{x \to 1} \frac{x^4-x^3+x^2-3x+2}{x^3-x^2-x+1}= \lim_{x \to 1} \frac{x^4-x^3+x^2-x-2x+2}{x^3-x^2-x+1}=\\=\lim_{x \to 1} \frac{x^3(x-1)+x(x-1)-2(x-1)}{x^2(x-1)-(x-1)}=\lim_{x \to 1} \frac{(x-1)(x^3+x-2)}{(x-1)(x^2-1)}=\lim_{x \to 1} \frac{x^3+x-2}{x^2-1}=\\=\lim_{x \to 1} \frac{x^3+x-2}{x^2-1}=\lim_{x \to 1} \frac{x^3-x+2x-2}{x^2-1}=\lim_{x \to 1} \frac{(x-1)(x^2+x)+2(x-1)}{x^2-1}=\\=\lim_{x \to 1} \frac{(x-1)(x^2+x+2)}{(x-1)(x+1)}=\lim_{x \to 1} \frac{x^2+x+2}{x+1}=\frac{1^2+1+2}{1+1}=\frac{4}{2}=2[/tex]
Przykład 4.
[tex]\lim_{x \to 3} \frac{2x^3-11x^2+12x+9}{x^4-6x^3+13x^2-24x+36}=\lim_{x \to 3} \frac{2x^3-12x^2+18x+x^2-6x+9}{x^4-6x^3+9x^2+4x^2-24x+36}=\\=\lim_{x \to 3} \frac{2x(x^2-6x+9)+(x^2-6x+9)}{x^2(x^2-6x+9)+4(x^2-6x+9)}=\lim_{x \to 3} \frac{(x^2-6x+9)(2x+1)}{(x^2-6x+9)(x^2+4)}=\\=\lim_{x \to 3} \frac{2x+1}{x^2+4}=\frac{2*3+1}{3^2+4}=\frac{7}{13}[/tex]
