homework help

Homework policy and details how to submit homework

Submissions without explanations will not earn any credit. A so-

lution is a process and steps illustrating how you arrive at a final

answer and at a conclusion. We grade the logical process - not the

answers which you can check on the homework forum.

Note when you solve limit

problems, do not forget to

write lim each time. For

example:

wrong :

lim x→2(x

2 + x)

= x2 + x = 22 + 2 = 6.

correct :

lim x→2(x

2 + x)

= lim x→2(x

2) + lim x→2 x

= (lim x→2 x)

2 + lim x→2 x

= 22 + 2 = 6.

1. Find the limits (a) lim x→0− 2x, (b) limx→3+ x, (c) limx→3+ 5, (d) limx→−1− |x+

2|? In each case if you use a limit law, mention which one you are using.

2. Sketch by hand the graph of the function

f(x) =

{

1

2 x if x < −1

x if x ≥ −1

Find lim x→−1− f(x). Find limx→−1+ f(x). Are these two limits the

same? What can you say about lim x→−1 f(x)?

3. Use the Limit Laws to find the limit. Graph each function before you

submit the solution to check if the graph supports your conclusion. No

need to include the graph though. You could use the grapher linked from

the top right corner of this page. lim x→3(x

2 − x + 2).

4. Use the Limit Laws to find the limit lim x→3(x

2 − x + 2/x3). [Note:

x2 − x + 2/x3 6= x 2−x+2 x 3 .

5. Use the Limit Laws to find the limit lim x→−3

x 2−9 x+3

.

6. Use the Limit Laws to find the limit lim x→7

√ x−

√ 7

x−7

7. Use the Limit Laws to find the limit lim x→5

x 3−125 x−5

8. Does the limit lim x→1

x 2−x−1 x−1

exist?

9. Use Limit Laws to find the limit lim x→1

x 2 +x−2 x−1

.

10. Graph the function f(x) =

{

sin π x

if x 6= 0

0 if x = 0 where −0.2 ≤ x ≤ 0.2.

Then graph f(x) where x varies over a smaller interval, for example,

−.06 ≤ x ≤ 0.06. Determine lim x→0 sin

π

x if it exists. Describe in a

paragraph the behavior of the values of f(x) near x = 0. Include graphs

in your solutions.

11. Graph the function

f(x) =

{

x2 sin π x

if x 6= 0

0 if x = 0 where − 0.2 ≤ x ≤ 0.2.

Then graph f(x) where x varies over a smaller interval, for example,

−.08 ≤ x ≤ 0.08. Describe in a paragraph the behavior of the values of

f(x) near x = 0. The video on the margin might help. Based on this

video what are the two functions g(x) and h(x) squeezing f(x), that

is such that h(x) ≤ f(x) ≤ g(x)? What is lim x→0 h(x)? and what is

lim x→0 g(x)? Determine limx→0 x

2 sin π x if it exists. Include graphs in

your solutions.

12. Discuss on the forum but do not submit. Let f(x) be the smallest digit

that appears in the decimal expansion of x. For example, f(0.123123123123 . . .) = 1

2

0 and f(1.1111 . . .) = 1, and f(2.10101010101 . . .) = 0. Find if they

exist these limits: (a) lim x→1 f(x), (b) limx→2 f(x)?

13. Discuss on the forum but do not submit. We define f on a colored

interval [0, 1).

f(x) =

{

x if x is blue

1 otherwise.

(a) Determine lim x→1− f(x) in the case when the entire interval [0, 1)

is blue? (b) Determine lim x→1− f(x) in the case when the entire in-

terval [0, 1) is black? (c) Does there exist lim x→1− f(x) if near 1 (in

every neighborhood of 1) there are numbers colored blue and there are

numbers colored black? Hand sketch a possible graph of f.