AP Calculus BC
Mini-Project 1 : Vertical and Horizontal Asymptotes and Other Discontinuities
Name:___________________________
In this mini-project, you will create a piecewise function that has the characteristics listed below. You
will write the piecewise function algebraically, and you will graph the piecewise function (using
WinPlot). SHOW ALL WORK (i.e., explain WHY your function satisfies the required conditions).
Step 1: Pick five (5) numbers (any real numbers, positive or negative, but all must be different and at
least one must be negative):
A: ______
B:_______
C:_________
D:________
E:_________
Step 2: Create a piecewise function that has:
a) A hole at x = A.
b) A vertical asymptote at x = B
c) A jump at x = C
d) A “breakpoint” in the piecewise function at x = D, but lim f ( x) exists. (A “breakpoint” is a
x D
value of x where the domain of the piecewise function “breaks” into its pieces.)
e) A horizontal asymptote at y = E.
Write your piecewise function here.
On a separate sheet of paper, print out your function and attach it to this document.
Step 1: Using Limits, demonstrate why your piecewise function has a hole at x = A. Explain why this is a
removable discontinuity and how you would remove it.
Step 2: Using the limit definition of VA, demonstrate why your function has a vertical asymptote at x =
B. Are there any other vertical asymptotes for your function? Explain why or why not.
Step 3: Evaluate lim f ( x) .
x B
Step 4: Using limits, demonstrate why your function has a jump at x = C. Explain, using the
mathematical definition of continuity why your function is NOT continuous at x = c and why that
discontinuity is non-removable.
Step 4: Using limits demonstrate that the limit of f(x) as x approaches D exists. Is your function
continuous at x = D? Explain why or why not, using the mathematical definition of continuity.
Step 5: Using the limit definition of HA, demonstrate why your function has a horizontal asymptote at y
= E. Is there a value of x such that f(x) = E anywhere in your domain? Explain why or why not. Are
there any other horizontal asymptotes for your function? Explain why or why not.
Step 6: Identify any other points of discontinuity in your function. For each point of discontinuity,
specify whether the discontinuity is removable or nonremovable. If the discontinuity is removable, find
the value that removes the discontinuity.
Step 7: Pick an interval on your domain where the IVT applies. Explain why the IVT applies. Draw SOME
conclusion about your function using the IVT.