Mathematics, 28.05.2020 22:57 tantoine2704
Use the limit theorem and the properties of limits to find the horizontal asymptotes of the graph of the function f(x)= 3x^2+5 / 4x^2-6x+2
Answers: 3
, and L is finite
![\lim_{x\rightarrow \infty^- }\left [ \frac{3+\frac{5}{x^2}}{4-\frac{6}{x}+\frac{2}{x^2}} \right ]](/tpl/images/0669/2898/4ad33.png)
![=\frac{\lim_{x\rightarrow \infty^- } \left [ 3+\frac{5}{x^2} \right ]}{\lim_{x\rightarrow \infty^- }\left [ 4-\frac{6}{x}+\frac{2}{x^2} \right ]}\\=\frac{3+0}{4-0+0}\\=\frac{3}{4}](/tpl/images/0669/2898/077c6.png)
![\lim_{x\rightarrow \infty^+ }\left [ \frac{3+\frac{5}{x^2}}{4-\frac{6}{x}+\frac{2}{x^2}} \right ]=\frac{\lim_{x\rightarrow \infty^+ } \left [ 3+\frac{5}{x^2} \right ]}{\lim_{x\rightarrow \infty^+ }\left [ 4-\frac{6}{x}+\frac{2}{x^2} \right ]}\\=\frac{3+0}{4-0+0}\\=\frac{3}{4}](/tpl/images/0669/2898/cc139.png)
![\lim_{x\rightarrow \infty^- }\left [ \frac{3+\frac{5}{x^2}}{4-\frac{6}{x}+\frac{2}{x^2}} \right ]=\lim_{x\rightarrow \infty^+ }\left [ \frac{3+\frac{5}{x^2}}{4-\frac{6}{x}+\frac{2}{x^2}} \right ]=\frac{3}{4}](/tpl/images/0669/2898/9d9ea.png)
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