fkb20203-t ts and continuity)
DESCRIPTION
ZCXSAFASDFTRANSCRIPT
-
TUTORIAL 2
Engineering Technology
Mathematics 2 Limits and Continuity
Universiti Kuala Lumpur Malaysia France Institute, Technical Foundation,
Mathematics Unit, [email protected]
The surest way not to fail is to determine to succeed
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
1
1. Refer to the graph and answer questions (a) (c)
(a) at 2x = : 2)f(andf(x)lim,f(x)lim,f(x)lim-2x-2x-2x
+
(b) at 1x = : f(1)andf(x)lim,f(x)lim,f(x)lim1x1x1x +
(c) at 3x = : f(3)andf(x)lim,f(x)lim,f(x)lim3x3x3x +
2. Refer to the graph and answer questions (a) (f)
(a) Determine
(i) f(0)
(ii) f(1)
(iii) f(2)
(iv) f(3)
(v) f(4)
(b) What is the limit of f(x) as x approaches 0 from the left?
(c) What is the limit of f(x) as x approaches 1 from the left?
(d) Does the limit exist as x approaches 1 from both sides? Explain why.
(e) What is the limit of f(x) as x approaches 2 from the left? The right?
(f) What is the limit of f(x) as x approaches 2 from both sides? Explain why.
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1 0 1 2 3 4
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
2
3. Sketch the graph of a function f that satisfies all of the following conditions:
(a) Its domain is the interval [ 0 , 4 ]
(b) 1f(4)f(3)f(2)f(1)f(0) =====
(c) 2f(x)lim1x
=
(d) 1f(x)lim2x
=
(e) 2f(x)lim3x
=
4. In the exercise below the graph of a function is given. State whether or not f(x)lim3x
exists, and if it does, give its value.
5. Sketch the graph of the piecewise function below and then find each of the following
limits or function values. If the limit does not exist, write DNE, if the value is not defined
write undefined.
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
3
6. Use the Principal Limit Theorem to find each of the following limits. Justify each step with a named rule.
(a)
3x
3xlim
4x (b)
2x
3lim
2x
(c)
3x
27xlim
3
3x (d)
25x
5xxlim
2
2
5x
(e)
++
+
1917x3xx
76x2xxlim
4659
185689
1x (f) elim
4x
(g)
1x
1)(xlim
4
4
1x (h)
4x
1lim
2x
2
2x
(i)
+
++
43xx
3627x13x3xxlim
2
234
1x (j)
4
2
3x x
x12lim
(k)
+
x
33xlim
0x (l)
x
x255lim
0x
7. Find the limits if 2f(x)limCx
=
and 3g(x)limCx
=
(a) 4g(x)2f(x)limCx
(b) ( )3Cx
1f(x)lim +
(c)
+
f(x)g(x)
3g(x)2f(x)lim
Cx (d) ( )2g(x)f(x)lim
Cx
(e) ( )3g(x)f(x)limCx
+
(f)
g(x)1
f(x)lim
2
Cx
8. Find the numbers a and b such that : 1x
2baxlim
0x=
+
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
4
9. For
+
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
5
(e)
cosx1
sinxxlim
0x (f) )( cosxelim
x
0x
(g)
x2x
sinxlim
20x (h)
x
xsinlim
2
0x
14. Determine the vertical and horizontal asymptotes (if any) of the following functions
(a) 5x
2f(x)
=
(b) 2x4
xf(x)
=
(c) 64xx
13xf(x)
2
2
++
+=
(d) 3xx
43xf(x)
2
=
(e) 32x
17xf(x)
=
(f) 25x
1xf(x)
2
=
15. Given 5lnx4xf(x)5xx2
+
for all x , find f(x)lim1x
16. Find f(x)lim0x
when 1xf(x)cosx2
+
for all 1x1
17. If 3
3
x
73xf(x)
x
23x +
for all 20x > , find f(x)limx
18. Find the following limits by using the Sandwich/Squeeze Theorem
(a)
x
sinxlimx
(b)
3
2
x 5x
sin3xxlim
(c)
2
2
x 5x
sin3xxlim
(d)
+
+
4x
5x2cosxlim
2
2
x
19. Use the Sandwich/Squeeze Theorem to show that 0x sinxxlim 23
0x=
+
20. Prove that 0x
2 cosxlim 4
0x=
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
6
21. Determine the discontinuity of the function 3,2xatf(x) =
22. Determine the discontinuity of the function 2,1xatf(x) =
23. Refer to the following figure.
Explain why the function f is not continuous at 1xand1x,3x ===
0
1
2
3
4
5
6
7
8
9
10
-3 -2 -1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1 0 1 2 3 4
y
x
-4 -3 -2 -1 1 2
4
2
-
Engineering Technology Mathematics 2 FKB20203
ET
M2
- L
imit
s a
nd
Co
nti
nu
ity
7
24. Find the values of dandc that make h(x) continuous on
>
+