Tutorial 6 - Limit
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Limits of a Function
1. In problem a b, use your calculator to complete the table,
and use your
results to estimate the given limit.
a) x
ex
x
1
lim0
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
b)2
4lim
2
2 +
x
x
h
x -2.1 -2.01 -2.001 -1.999 -1.99 -1.9
f(x)
2. In problem below, use the graph of f to estimate each limits
if it exists.i) a) )(lim 1
xfx b)
)(lim1
xfx c)
)(lim2
xfx
y
3
2
y =1
x
-1 1 2
ii) a) )(lim 1xf
x b))(lim
0xf
x c))(lim
1xf
x
y
1
- 1 1 x
- 1
iii) a) )(lim1
xfx
b) )(lim1
xfx +
c) )(lim1
xfx
1
Tutorial 6: QQM 1023
f(x)
f(x)
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8/3/2019 Tutorial 6 - Limit
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d))(lim
2
xfx
e))(lim
2
xfx
+
f)
)(lim2
xfx
g))(lim
1
xfx
h))(lim
1
xfx +
i)
)(lim1
xfx
j) )(lim xfx k))(lim xf
x
f(x)
3
2
1
x
-2 -1 1
3. In problem below, find the limits. If the limit does not
exist, so state that.a) 16lim100x b)
)5(lim 25
tt
c) 5lim2
4++
ppp
d)1
1lim
1 +
+
x
x
xe)
x
xx
x
2lim
2
0
f)43
209lim
2
2
4
+
xx
xx
x
g)2
4lim
2
2
t
t
th)
1
1lim
41
+
h
h
hi) )2(lim
3
+
xx
j) xx5lim
k) 4
0
6lim
x
x
x l)
xxx 2
7lim
m)
x
x
x
23
42lim
n)
175
432lim
2
2
+
+
ww
ww
wo)
34
1lim
3
2
+
xx
x
x
p)74
125lim
2
+
++
t
tt
tq)
+
+ 1
11lim
1 xx
r))(lim,)(lim,)(lim
222
xfxfxfxxx +
if
>
=
21
22)(
x
xxf
s) Refer problem (r), )(lim,)(lim xfxf xx
2
Tutorial 6: QQM 1023
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8/3/2019 Tutorial 6 - Limit
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4. The maximum theoretical efficiency of a power plant is given
by
h
ch
T
TTE
=
where Th and Tc are the respective absolute temperatures of the
hotter and
colder reservoirs. Find;
a) EcT 0lim
b) Ehc TT
lim
3
Tutorial 6: QQM 1023
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Answers: Limits of a function
1. a)x
ex
x
1lim
0
= 1 b)2
4lim
2
2 +
x
x
h= - 4
2. i) a) 1 b) not exist c) 3
ii) a) 1 b) not exist c) 1
iii) a) 2 b) 3 c) not exist
d) + e) + f)
g) 1 h) 1 i) 1
j) k) 0
3. a) 16 b) 20 c) 5d) 1 e) -2 f) -1/5
g) 4 h) -1/4 i) 1
j) - k) - l) 0
m) -1 n) 2/5 o) 0
p) q)
r) )(lim,2)(lim,1)(lim 222xfxfxf
xxx ==
+ not exist
s) 2)(lim,1)(lim ==
xfxfxx
4. a) 1 b) 0
4
Tutorial 6: QQM 1023
