Math 137 Online Week 5 Outline and Practice Problems
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Math 137 Online
Week 5 Outline and Practice Problems
This week we’ll talk about infinite limits and continuity.
Reading & Videos
These are meant as “another look” of the content covered in class. It is not required that you read the notes or watch the videos but you should, in principle, know the material within after attending class. It is strongly recommended that you do go through this if you have the time. Also, there are more examples!
● (Course notes available here)
● Section 2.7.1 - Horizontal Asymptotes and Limits at Infinity (part 1), (part 2)
● Section 2.7.2 - Fundamental Log Limit
● Section 2.7.3 - Vertical Asymptotes and Infinite Limits
● Section 2.8 - Continuity
● Section 2.8.1 - Types of Discontinuities
● Section 2.8.2 - Continuity of Polynomials, Trig Functions and Exponentials
● Section 2.8.3 - Arithmetic Rules for Continuity
● Section 2.8.4 - Continuity on an Interval
Practice Problems
1. Compute the following limits using the fact that lim = 0 and lim = 0 for any p > 0.
^x + ln(x) · x
北→o 1 · ln(e2北)
(b) lim e-北 (1 · x^e北 )
北→o
(c) lim ln(北北) for p > 0.
北→o
(d) lim
2. Find all asymptotes (both horizontal and vertical) of f (x) =
1 |
·2x2 + 2 . |
3. Prove that the function f (x) = 2x2 + 9 is continuous at x = 2 using the ∈ · δ definition of continuity (the Formal Definition of Continuity II).
4. Let f be a function defined as
f (x) = ,0(北)北(-)4-
Find the intervals where f is continuous. Justify your answer.
5. Let
f (x) =
if x > 2,
if x < 2.
Find the value c such that f (x) is continuous on R. Justify your answer.
6. Show that if a function is continuous at x = 0 and satisfies:
(a) f (x + y) = f (x) + f (y) then it is continuous everywhere
(b) f (x + y) = f (x)f (y) then it is continuous everywhere
┌Hint: You may use the fact that l北f (x) = f (a) is equivalent to h(l)f (a + h) = f (a).┐
Practice Quiz
1. Over which intervals is the function f (x) = ^2x · x2 + 8 continuous:
(a) [ ·2, 4]
(b) [ ·4, 2]
(c) ( ·4, 2)
(d) ( ·2, 5)
(e) ( ·2, 4)
(f) [4, o)
(g) (0, 1]
(h) ( ·o, ·2)
(i) None of the above.
2. Suppose that f (x) is a function such that lim f (x) = 4 and lim f (x) = o. Which of the following 北→o 北→ -o
might be true?
(a) f (x) is continuous at each a e R
(b) lim f (x) = ·o
北→7 −
(c) f (0) = 4
(d) f (x) < 0 for at least 13 values of x
(e) For all x e R we have that f (x + 4) = f (x).
(f) None of the above.
2023-02-13