Question

f(x)=(3(x^2)-1)/(4-(x^2))
find horizontal and vertical asymptotes with limits

Answer 1

To find the horizontal symptote(s), find the limit of the function as x ------> infinity

Answer 2

As x --------> infinity, the function approaches -3, so y = -3 is a horizontal asymptote.

Answer 3

For the vertical asymptotes, set denominator equal to zero, x = 2 and x = -2.

Answer 4

i understood for the vertical aysmptotes that it was 2 and -2 by simply factoring, however when i looked at the solution, i didnt understand this!

\lim_{x \rightarrow 2^{-}} f(x) =\left(\begin{matrix}11 \\ 0^{+}\end{matrix}\right) = +infty

Answer 5

\[\lim_{x \rightarrow 2^{-}} f(x) =\frac{ 11 }{ 0^{+} } = +\infty\]

Answer 6

\[\lim_{x \rightarrow -2^{-}} f(x) =\frac{ 11 }{ 0^{-} } = - \infty\]

Answer 7

my question is, x is approaching from the left side of both -2 and 2, by why is one negative infinity and one is positive infinity??

Answer 8

You can plug in numbers (like -2.001 for a number left of -2, to see why, or 1.99 for a number left of 2)

or graph the function to see it's behaviour near asymptotes http://www.wolframalpha.com/input/?i=%283x%5E2-1%29%2F%284-x%5E2%29

Answer 9

ohh ok thank you!!