For each limit statement in Exercises , use algebra to find or in terms of or , according to the appropriate formal limit definition.
, find in terms of .
For , .
Now for .
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 of has a horizontal asymptote at .
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