What is the Method for Determining Infinity Limits in a Rational Function?

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ok I am confused when x->negative infinity or positive infinity.

for example

lim (5x^3+27)/(20x^2 + 10x + 9)
x-> negative infinty

heres what i think, i want to know if i have the right idea or not.

- so since the top exponent is larger then the denominator the lim DNE and so i plugged in a negative value to test if it is negative or positive infinity so i put in -1 just to test, the thing is i only plunged it into the (5x^3)/(20x^2); I am thinking this is what i do, and then i get a negative value so i am assuming it is negative infinity. that is what the answer is so posed to be, but did i do it how its so posed to be done?
 
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Divide top and bottom by x^2 and get rid of stuff that tends to zero.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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