WRITTEN ASSIGNMENT 9
Hand in full solutions to the questions below. Make sure you justify all your work and include complete
arguments and explanations. Your asnwers must be clear and neatly written, as well as legible (no tiny
drawings or micro-handwriting please!). Your answers must be stapled, with your name and student number
at the top of each page.
1. Sketch the graph of a function f that satisfies all of the given conditions:
• f (0) = 2, f (1) = 0
• lim f (x) = −∞
x→3
• lim f (x) = 2
x→∞
•
lim f (x) = 2
x→−∞
2. Sketch the graph of a function g that satisfies all of the given conditions:
• g(1) = 0
• lim g(x) = −∞
x→0
• lim g(x) = 2
x→∞
•
lim g(x) = −2
x→−∞
• g 0 < 0 if x < 0 and x > 2, and g 0 > 0 if 0 < x < 2
Explain why g(2) > 2 and why g must have an inflection point at x = a with a > 2.
3. Find numbers a and b such that
√
lim
x→0
ax + b − 2
=1
x
4. Find the equation of the horizontal and vertical asymptotes, if they exist, of the function f (x) =
and explain why they are asymptotes of f . If no such asymptote exists, explain why.
esin x
x
5. (a) Explain why the following calculation is wrong.
lim
x→π −
sin x
cos x
= lim
= −∞
−
1 − cos x x→π sin x
(b) Below is an incomplete calculation. Explain how the second limit was obtained, and why one is
allowed to equate the two limits. Then complete the calculation and evaluate the limit.
cos x − 1
sin x − x
= = lim+
=
lim+
x
sin
x
sin
x + x cos x
x→0
x→0
1