Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 8
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41Worksheet # 6: Algebraic Evaluation of Limits, Trigonometric Limits
1. For each limit, evaluate the limit or or explain why it does not exist. Use the limit rules to justify each
step. It is good practice to sketch a graph to check your answers.
+ 2
1
3
(a) lim
(d) lim
2
4
2
2
2
2
2
2
9
(b) lim
(e) lim
2
4
3
2
9
3
2
(2 + )
8
(c) lim
(f) lim
2
+ 4
2
0
2. Let ( ) =
( + )
( )
(a) Let ( ) = lim
and find ( ).
0
(b) What is the geometric meaning of (4)?
(c) What is the domain of ( )?
2
3. Let ( ) = 1+
sin(1
) for
= 0. Find two simpler functions
and
so that we can use the squeeze
theorem to show lim
( ) = lim
( ) = lim
( ). Give the common value of the limits.
0
0
0
Use your calculator to produce a graph that illustrates that the squeeze theorem applies.
4. Evaluate the limits
sin(2 )
1
cos(2 )
(a) lim
(e) lim
0
0
tan(2 )
1
cos(2 )
(b) lim
1+cos(2 )
(f) lim
: Multiply by
.
0
2
1+cos(2 )
sin
(2 )
0
cos( ) tan(2 )
(c) lim
0
1
cos( )
(g) lim
(d) lim
csc(3 ) tan(2 )
2
0
The following identity may be useful for the next problems.
cos( + ) = cos( ) cos( )
sin( ) sin( )
(1)
5. Use equation (1), to simplify the limit
cos( + )
cos( )
lim
0
6. Evaluate the following limits
2
1
cos(3 )
(e) lim
Multiply and divide
(a) lim
2
sin
0
0
by 1 + cos(3 )
2
cos(5 )
cos
(5 )
(b) lim
cos
cos(4 )
0
(f) lim
Use equation (1)
tan(11 )
2
0
(c) lim
to rewrite cos(4 ) as cos( + 3 )
5
0
cos(2 )
1
(d) lim
Use equation (1)
cos( )
1
0
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