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Example 1 : Find the limit of $\lim_{x \rightarrow \infty}\frac{2x-1}{3x +5}$
Solution : $\lim_{x \rightarrow \infty}\frac{2x-1}{3x +5}$
Take x as a common factor from numerator and denominator
= $\lim_{x \rightarrow \infty}\frac{x(2-\frac{1}{x})}{x(3 +\frac{5}{x})}$
= $\lim_{x \rightarrow \infty}\frac{2-\frac{1}{x}}{3 +\frac{5}{x}}$
( $\lim_{x \rightarrow \infty}\frac{1}{x} = 0)$
$\lim_{x \rightarrow \infty}\frac{2x-1}{3x +5} = \frac{2}{3}$
Example 2 : Find the limit of $\lim_{x \rightarrow \infty}\frac{2x^{2}+4 }{x^{2} -5x -1}$
Solution : $\lim_{x \rightarrow \infty}\frac{2x^{2}+4 }{x^{2} -5x -1}$
Take $x^{2}$ as a common factor from numerator and denominator
= $\lim_{x \rightarrow \infty}\frac{x^{2}(2+\frac{4}{x^{2}})}{x^{2}(1 -\frac{5}{x}-\frac{1}{x^{2}})}$
$\lim_{x \rightarrow \infty}\frac{(2+\frac{4}{x^{2}})}{(1 -\frac{5}{x}-\frac{1}{x^{2}})}$
( $\lim_{x \rightarrow \infty}\frac{1}{x} = 0 $ and $\lim_{x \rightarrow \infty}\frac{1}{x^{2}} = 0$)
$\lim_{x \rightarrow \infty}\frac{2x^{2}+4 }{x^{2} -5x -1}$ =2
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