Proof (a) Prove that if lim x → c | f ( x ) | = 0 , then lim x → c f ( x ) = 0 . ( Note: This is the converse of Exercise 114.) (b) Prove that if lim x → c f ( x ) = L , then lim x → c | f ( x ) | = | L | . [ Hint: Use the inequality ‖ f ( x ) | − | L ‖ ≤ | f ( x ) − L | .]
Proof (a) Prove that if lim x → c | f ( x ) | = 0 , then lim x → c f ( x ) = 0 . ( Note: This is the converse of Exercise 114.) (b) Prove that if lim x → c f ( x ) = L , then lim x → c | f ( x ) | = | L | . [ Hint: Use the inequality ‖ f ( x ) | − | L ‖ ≤ | f ( x ) − L | .]
determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If
lim x→7 g(x) = 0
and
lim x→7 f(x)/g(x) = 0
exists, then
lim x→7 f(x) = 0.
True. If lim x→7 f(x) is not equal to zero and lim x→7 g(x) = 0, then lim x→7 f(x)/g(x) does not exist.
True. Any number divided by zero is equal to zero.
False. Let g(x) = (x − 7) and f(x) = (x − 1)(x − 7). Then lim x→7 g(x) = 0 and lim x→7 f(x)/g(x) = 0 exists, but lim x→7 f(x) is not equal to 0.
False. Divison by zero is not allowed.
False. There is not enough information given to determine lim x→7 f(x)/g(x)
lim x→∞
a) value of f(1)
b) lim x→1-f(x)
c) lim x→1+f(x)
d) Does lim x→1 f(x) exist? If so, find value. If not, explain why.
e) lim x→2+f(x)
f) lim x→2-f(x)
Chapter 2 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
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