4.3- Vertical and Horizontal Asymptotes

Dividing 2 polynomial functions results in

function that is not a polynomial - quotient is a rational function

Rational Functions:

function that is a fraction and both numerator and denominator are polynomials

Asymptote:

special features and play big role in curve sketching

Vertical Asymptotes:

when is denominator equal to 0

F(x)= p(x)/q(x) has vertical asymptote at

x=c IF q(c) = 0 BUT p(c) cannot equal 0 (c must not give a numerator of 0)

If x value does give numerator of zero when looking for Vertical asymptote:

results in indeterminate case

When evaluating behaviour as it gets close to infinity (horizontal asymptote ):

• When large negative value is subbed in, you get a negative number or number below Horizontal asymptote (approaching from bottom)

• When large positive value subbed in, you will get a small positive value or number slightly bigger than horizontal asymptote (approaching from top)

2 values of limits:

• List of simple limits

• Rewrite in terms of highest degree

List of simple limits:

• substitute big positive and negative values

• if variable on top or big number/very big number (higher degree) limit will always = 0

Rewrite in terms of highest degrees

• a polynomial can always be written so term of highest degree is a factor with coefficient positive or negative

When examining if function approaching from above or below horizontal axis that is not 0:

do not only consider sign, consider if VALUE is less or more than horizontal asymptote

Analysis for horizontal asymptote has three columns:

value of x (approach value from positive or negative side), f(x) (sign) , f(x)→ (positive or negative infinity)

When pulling out coefficient in front of limit when calculating horizontal asymptote do not pull out:

variable because that is what is being evaluated

Lim x→ -infinity F(x) → - means :

approaches from below

Lim x→ +infinity F(x) → + means:

approaches from above