Question

Please help me. Describe these long term behavior.​

Answer 1

Check the picture below.

recall the oblique asymptote occurs when the degree of the numerator is greater by 1, in these two cases, that's what happens.

notice the quotient on the first one of x - 2, namely the oblique asymptote is y = 1x - 2, so its slope is well, you already know, is 1.

on the second one in the picture below, the quotient is 3x - 3, namely the oblique asymptote is y = 3x - 3, so the slope is just as you guessed it, is 3.

Answer 2

1. b. slant asymptote with a slope of 1
2. b. slant asymptote with a slope of 3

Explanation:

When the rational function has a numerator polynomial that has a degree greater than or equal to the degree of the denominator polynomial, you can find the asymptote by performing polynomial long division and looking at the quotient.

1. The quotient of the division is x -2, so the slant asymptote is the line ...

y = x -2

This line has a slope of 1, matching choice B.

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2. Likewise, the quotient from the division of these polynomials is 3x -3, so the slant asymptote is ...

y = 3x -3

This line has a slope of 3, matching choice B.

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Comment on asymptotes

We usually study functions with vertical, horizontal, or slant asymptotes. The asymptotes don't need to be restricted to these. For any rational function, the asymptotic behavior will match the quotient of the division. (You can have parabolic or cubic asymptotic behavior, for example.) It is only the remainder or fraction portion of the division result that goes to zero when the variable gets large.