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A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists.
Show that each of the following functions has a horizontal asymptote by calculating the given limit.
$\displaystyle{ \lim_{x\to\infty} 4+\frac{4 x}{x^2-15 x +5}= }$
$\displaystyle{ \lim_{x\to-\infty} \frac{7-7 x}{12+x}+\frac{4 x^2 +5}{(9 x-13)^2}= }$
$\displaystyle{ \lim_{x\to-\infty} \frac{10 x+13}{x-9}\cdot\frac{12 x-4}{-x-10}= }$
$\displaystyle{ \lim_{x\to\infty} \sqrt{x^2+12 x -10}-x= }$
$\displaystyle{ \lim_{x\to-\infty} \sqrt{x^2+12 x-10}+x= }$

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