Example: Evaluate limx→−∞x+3√9x2−5x.
Solution:
As x→−∞, the leading term at the numerator is x. At the denominator, on the other hand, (9x2−5x)∼9x2, so the denominator approaches √9x2=3|x|. Since x→−∞, it's safe to assume that x<0, so that |x|=−x. All together,
x+3√9x2−5x∼x−3x→−13,
which is the value of the limit. In partticular, this shows that y=−1/3 is a horizontal asymptote.
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