Question

Determine all horizontal, slant, and vertical asymptotes. For each vertical asymptote, determine whether f(x) -> infinity sign of f(x) -> negative infinity sign on either side of the asymptote.

Answer 1

f(x) = x^2/4 - x^2

Answer 2

is it
\[\frac{x^2}{4-x^2}\]?

Answer 3

yes

Answer 4

ok
so for the vertical asymptotes, set the denominator equal to zero and solve for \(x\)

Answer 5

ok

Answer 6

you get
\[4-x^2=0\] or \[(2-x)(2+x)=0\] and the solutions are
\(x=-2\) or \(x=2\)
those are your vertical asymptotes (there are two of them)

Answer 7

ok

Answer 8

for the horizontal asymptote, note that the numerator and denominator have the same degree (both are degree 2) so it is the ratio of the leading coefficients

Answer 9

okay

Answer 10

the leading coefficient of \(x^2\) is 1 and the leading coefficient of \(4-x^2\) is \(-1\)
therefore the horizontal asymptote is \(y=-1\)

Answer 11

that is in the case where the degrees are the same. there is no slant asymptote

for there to be a slant asymptote , the degree of the numerator would have to be one more than the degree of the denominator

Answer 12

okay thx

Answer 13

yw