MATH 100 V02 — Homework 4: written component
Mihai Marian
Due at 8:00 am on Thursday, February 10.
Answer the following questions using pencil and paper. Your answers will be evaluated on their correctness as well as their clarity.
1. (Asymptotes)
(a) (A warm-up) Consider the function f defined by f (x) =
1
(x+3)2
+ 1.
i. What is the domain of f ?
ii. What is the vertical asymptote of f ? (Recall that an asymptote is a line, so be
explicit about how your answer describes a line).
iii. Verify that the line above is indeed a vertical asymptote (by taking a limit).
iv. Verify the appropriate limits to find the horizontal asymptote of f .
v. Draw a well-labelled graph of f (a well-labelled graph needs to have labels on the
following: axes, x-intercepts, y-intercept and asymptotes).
√
(b) (As hard as it gets) Consider now the function g defined by g(x) =
as above:
x6 +x4
.
x(1−x2 )
Do the same
i. What is the domain of g?
ii. List the three values a for which {x = a} could be a vertical asymptote of g.
iii. Verify that one of those√is not a vertical√asymptote (Hint: you may want to use the
algebraic manipulation x6 + x4 = |x| · x4 + x2 and to compute one-sided limits).
iv. Verify
that the
other two are vertical asymptotes. (Hint: do an algebraic step such
√
√
x6 +x4
x6 +x4
1
as x(1−x2 ) = x(1−x) · 1+x
and use Theorem 1.5.9 ).
v. What is (or are) the horizontal asymptote(s) of g? Compute the limit(s) to check.
vi. (Optional) Draw a well-labelled graph of g.
(c) How many vertical asymptotes and how many horizontal asymptotes can the graph of a
function have? Explain why in 1 or 2 lines.
cos x
2. Describe the points of discontinuity of the function 2 sin(x)−1
and classify them (as removable,
jump, infinite, or other). Explain in a few lines why you classified them the way you did.