For each of the following, determine the equations of any horizsontal asymptotes and state wheter the curve?

Question

For each of the following, determine the equations of any horizsontal asymptotes and state wheter the curve approaches the asymptote from above or below.

a) x / x + 4

b) f(x) = 2x / x2 - 1

c) g(t) = 3t2 + 4 / t2 - 1

d) y = 3x2 - 8x - 7 / x -4

Answer 1

a.) horz asym. = 1 (if leading variable have the same value in their exponents, divide the leading coefficients to find it.
x/x is (1)x/(1)x = 1/1 =1
approaches from both ways

b) horz asym = forgot the rules on this but I know it approaches 0 but the are three parts due to two vertical asymptotes and in the middle of those two, it passes through 0.... I think the rules of thumb was if exponent of the denominators variable was one higher than the one in the numerator, it was 0
approaches from both ways

c) horz = 3 (see a for explanation)
Approaches from above

d) no horizontal asymptote but an oblique one which would be 3x+4 (if numerators leading variables exponent is higher BY ONE from the denominators leading variables exponent, divide: numerator/denominator, ignore the remainder to find the oblique asymptote.
Approaches? I'd say not applicable for its oblique but I guess both, if I was guessing.

Just an add in, not sure how creationist got infinity but if you are able to graph out (3x^2-8x-7)/(x-4) and then graph 3x+4, you will see the line 3x+4 is clearly the oblique asymptote and no horizontal exists.

Answer 2

For the horizontal asymptote, take the take the variable with the highest power from the numerator and denominator and put them together alone and simplify it. If you are left with a high value in the numerator then the answer is infinity, if the higher value is in the denominator then its 0. If the "x"s cross out then the answer is whoever fraction you are left with, take the limit as the variable approaches infinity.

a) the variable on the top and bottom in just "x." So take x/x and would be equal to 1. So you answer is f(x)=1.

b) the highest variable in the top in "2x" and I'm going to guess that x2 means x^2 so the fraction you would use is "2x/x^2." Cancel out an "x" on the top and bottom and then you have "2/x." Since you are left with a variable on the bottom the answer is f(x)=0.

c) The highest value on the top is 3t^2 and the bottom is t^2. so use 3t^2/t^2. Cancel out t^2 on the top and bottom and you have 3/1. So the answer is f(x)=3

d)You have two variables on the top here. Since 3x^2 is greater than 8x, use 3x^2. and the bottom is just x. So you have 3x^2/x. Cross out an "x" from top and bottom and you have 3x/1. Since you are left with the x on top and no variable on bottom, you have f(x)=∞.(which means no asymptotes)

As for which side approaches, you have to either use a graphing devise, graph it by hand or draw it up in your head.