Question

Which of the following is false for f(x) = (10x^3-10x^2-10x)/(2x^5-2x)

Answer 1

1) Factor and simplify:

f(x) = [10x (x^2 - x - 1) ] / [ 2x (x^4 - 1) ] =5 (x^2 - x - 1) / (x^4 - 1)

2) Calculate limits

A) limit of f(x) when x -> 0 = 5(-1)/(-1) = 5

=> y-axis is not an asymptote and A is TRUE.

B) Lim of f(x) when x -> +/- ∞ =

5 * (x^2 / x^4 - x / x^4  - 1/ x^4 ) / (x^4 / x^4 - 1 /x^4) = 0 / ∞ = 0

=> x-axis is an asymptote and B is TRUE

C) Lim of f(x) when x -> - 1 =

5 * (x^2 - x - 1) / (x^4 - 1) = 5 * (1 + 1 - 1) / (1 - 1) = 5 / 0 = ∞

=> x = - 1 is an asymptote and C. is FALSE.

Now you have the answer.

If you want you can verify that the last option is TRUE.

I also enclose a picture showing the asymptotes which may help you.

Answer: Option C. is the false one.