Notes Review Prior to Section 1.4 Limits and Continuity
Review:
To find:
1.) HOLES – factor the numerator and denominator. Cancel any common factors.
When these canceled common factors equal zero, there is a hole.
2.) VERTICAL ASYMPTOTES – after you have found holes, set the simplified
denominator equal to zero. Any values where the remaining denominator can be
zero are vertical asymptotes. Always in the form: x = #.
3.) HORIZONTAL ASYMPTOTES – can only occur if the degree in the numerator
is less than or equal to the degree of the denominator. Find the highest power of
x. H.A is the number of these x’s in the numerator / the number of these x’s in
the denominator. Always in the form: y = #
4.) SLANT ASYMPTOTES – Also called OBLIQUE asymptotes. Occurs ONLY
when the degree of the numerator is EXACTLY one degree larger than the degree
of the denominator. To find SA, divide rational equation and delete the
remainder (the remainder will always approach zero). Always in the form:
y = mx + b.
Find all asymptotes and holes:
x 3 + 3x 2 − 4x + 1
1. y =
x2 − 2
x 2 + 4x − 6
2. y =
2x + 4
3. y =
x+2
x − 3x − 10
4. y =
2
x −1
x − 36
2
To sketch the graph of a rational function…
• Simplify f(x) if possible.
• Determine the H.A., V.A., and holes. Sketch them.
• Determine the slant asymptote, when appropriate.
• Determine the y-intercept by finding f(0). Plot the y-intercept.
• Determine the zeros of the function by setting the numerator equal to zero and
plot the x-intercepts.
• Determine additional points as needed to sketch a smooth curve. You may need
approximately 3 points per branch of the curve.
• Sketch a smooth curve, making sure to identify any discontinuities in the graph.
Ex1: f (x) =
x
x +1
6
4
2
-10
-5
5
-2
-4
-6
10
Ex2: f (x) =
x +1
x −1
6
4
2
-10
-5
5
10
5
10
-2
-4
-6
Ex3: f (x) =
x2
x−2
6
4
2
-10
-5
-2
-4
-6