176
Chapter 2 | Limits
2.3 EXERCISES In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s). 83. lim x →0 ⎛ ⎝ 4 x 2 −2 x +3 ⎞ ⎠
1 a + h −
1 a
lim h →0
, where a is a non-zero real-valued
98.
h
constant
sin θ tan θ
99. lim θ → π
x 3 +3 x 2 +5 4−7 x
84. lim x →1
x 3 −1 x 2 −1
100. lim x →1
x 2 −6 x +3
lim x →−2
85.
2 x 2 +3 x −2 2 x −1
lim x →1/2
101.
(9 x +1) 2
lim x →−1
86.
x +4−1 x +3
lim x →−3
102.
In the following exercises, use direct substitution to evaluate each limit. 87. lim x →7 x 2
In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example 2.23 to simplify the function to help determine the limit.
⎛ ⎝ 4 x 2 −1 ⎞ ⎠
lim x →−2
88.
2 x 2 +7 x −4 x 2 + x −2 2 x 2 +7 x −4 x 2 + x −2
lim x →−2 −
103.
1 1+sin x
89. lim x →0
lim x →−2 +
104.
2
e 2 x − x
90. lim x →2
2 x 2 +7 x −4 x 2 + x −2
lim x →1 −
105.
2−7 x x +6
91. lim x →1
106. 2 x 2 +7 x −4 x 2 + x −2 In the following exercises, lim x →1 +
ln e 3 x
92. lim x →3
assume that
In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit.
lim x →6 h ( x ) =6. Use these three facts and the limit laws to evaluate each limit. 107. lim x →6 2 f ( x ) g ( x ) f ( x ) =4, lim x →6 g ( x ) =9, and lim x →6
x 2 −16 x −4 x −2 x 2 −2 x 3 x −18 2 x −12
93. lim x →4
g ( x )−1 f ( x )
108. lim x →6
94. lim x →2
109. lim x →6 ⎛ 110. lim x →6 ⎛
⎞ ⎠
⎝ f ( x )+ 1 3
g ( x )
95. lim x →6
⎞ ⎠ 3
⎝ h ( x )
(1+ h ) 2 −1 h
96. lim h →0
2
g ( x )− f ( x )
111. lim x →6 112. lim x →6
t −9
97. lim t →9
t −3
x · h ( x )
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