Math 1310
0 - Engineering Calculus Ix
2
Quiz 2
4
Name:
September 5, 2014
Quiz Score:
/10
4. For the function f whose graph is given, state the value of each
Answer each question completely in the area below. Show all work and explain your reasoning. If
if it exists.
If it does
not exist,
explain
why.or notes are allowed.
the workquantity,
is at all ambiguous,
it is considered
incorrect.
No phones,
calculators,
Anyone found violating these rules will be asked to leave immediately. Point values are in the square
(a) lim question.
(b)are lim
(c) lim
f !x" If there
f !x"
f !x"
to the left ofxthe
l0
xany
l 3"other issues, please ask
x lthe
3! instructor.
(d)function
(e) isf !3"
lim f !x"
1. Consider the
g(x) whose graph
given by:
7–8 Ske
values o
xl3
y
7. f !x"
4
2
8. f !x"
0
2
4
x
5. For the function t whose graph is given, state the value of each
; 9–11 Us
limit, if
State the value of each below if it exists. If it does not, state why.
1
quantity, if it exists. If it does not exist, explain why.
(a) lim
xl0
(a)
lim
t!t"
(b)
lim
t!t"
(c)
lim
t!t"
Solution: We"can look at the value of !
g(x) as x gets really close to 0 from the left and it’s clear
(a) limx→0 g(x)
tl0
tl0
tl0
it approaches 3. From the right, g(x) also approaches 3. As discussed in class, the function g(x)
is “nice”
x = 0, meaning
can effectively
read lim
off thet!t"
value from the graph.
(d) around
lim t!t"
(e)youlim
t!t" just(f)
1
t l 2"
(b) limx→3− g(x)
t l 2!
tl2
9. f !x"
Solution: This notation means checking the value of g(x) as x gets close to 3 from the left.
FromGraphing
the graph, calculator
it’s clear thisor
value
is 4.
computer
with graphing software required
1. Homework
;
1
(c) limx→3+ g(x)
Solution: Similar to above, this notation means checking the value of g(x) as x gets close to 3
from the right. Here, that value is 2.
O1 penmirrors.com
(d) limx→3 g(x)
Solution: Notice that the above two limits differ, and for the limit as x → 3 to exist, we need
them to be the same. Thus, the limit does at not exist.
1
(e) g(3)
Solution: Note, even though the limit of the function as x → 3 does not exist, from the graph,
it’s clear g(3) does exist and corresponds to the shaded in circle which suggests g(3) = 3.
1/2
H
Quiz 2
Math 1310 - Engineering Calculus I
September 5, 2014
2. Compute the following limits if they exists. If they do not exist, state why.
2 1/2
(a) limx→3 x 2 + 3x − 5
Solution: As we discussed in class, polynomials are “nice” and don’t cause any problems as far as
limits (which you can convince yourself if you draw the picture). Thus, the limit as x to3 is just
the value of the function at x = 3, meaning we can simply plug in x = 3 to the above function and
we end up with (3)2 + 3 · 3 − 5 = 13.
2 1/2
(b) limx→0
1
x
Solution: Although it seems like many of you did this with a calculator, it’s easy to reason just
by considering the behavior of this function as x → 0. If x is positive, that is, approaching the
limit from the right, we see that 1/x is corresponds to dividing by a small positive number and is
therefore a huge positive number. Similarly, as x approaches 0 from the left, we’re dividing by a
small negative number, meaning we’re left with a huge negative number. By this reasoning, it’s
impossible for a huge negative number and a huge positive number to be equal, so the limit cannot
exist. You can see this from the graph below:
4
2
-4
-2
2
4
-2
-4
2/2