- #1
zooxanthellae
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Homework Statement
Prove that lim (1/x) x --> 0 does not exist, i.e., show that lim (1/x) x --> 0 = l is false for every number l.
Homework Equations
0 < |x-a| < d
|f(x) - l| < E
The Attempt at a Solution
The strange thing is, the first time through I got the same solution as Spivak, but looking over it again the logic seems downright wrong. Here's the solution, verbatim:
The function f(x) = 1/x cannot approach a limit at 0, since it becomes arbitrarily large near 0. In fact, no matter what d > 0 may be, there is some x satisfying 0 < |x| < d, but 1/x > |l| + E, namely, any x < min(d, 1/(|l| + E)). Any such x does not satisfy |(1/x) - l| < E.
Where is he getting the bold portion from?
I write |1/x - l| < E
|1/x| - |l| < |1/x - l| < E
|1/x| - |l| < E
|1/x| < E + |l|
How is he getting greater-than?