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Chapter 1: Q. 14 (page 153)
Consider the limit expression
Calculate the limit.
The limit of the expression is.
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Each function in Exercises 9鈥�12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.
Write a delta鈥揺psilon proof that proves that is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to .
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.
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| < .
Use what you know about one-sided limits to prove that a function is continuous at a point if and only if it is both left and right continuous at .
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