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Chapter 1: Q. 64 (page 108)
For each of the limit statements in Exercises 61-66, write a , or proof, according to the type of limit statement.
The proof of given limit is,
.
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Write delta-epsilon proofs for each of the limit statements in Exercises
For each limit statement , use algebra to find 未 > 0 in terms of > 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < .
Sketch a labeled graph of a function that satisfies the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem follows.
If is a continuous function, what can you say about
Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) A limit exists if there is some real number that it is equal to.
(b) The limit of as is the value .
(c) The limit of as might exist even if the value of does not.
(d) The two-sided limit of as exists if and only if the left and right limits of exists as .
(e) If the graph of has a vertical asymptote at , then .
(f) If , then the graph of has a vertical asymptote at .
(g) If , then the graph of has a horizontal asymptote at .
(h) If, then the graph ofhas a horizontal asymptote at.
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