- #1
farleyknight
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- 0
Homework Statement
Prove that [itex]\lim_{x \to 1} \frac{x}{x - 1} = \infty[/itex]
Homework Equations
The Attempt at a Solution
First [itex]0 < |x - 1| < \delta[/itex] implies [itex]\frac{x}{x - 1} > N, \forall N[/itex]
Because the reals are dense, we can choose an [itex]n > 0[/itex] such that [itex]\frac{x}{x - 1} > \frac{n}{x - 1} > N[/itex]
then [itex]\frac{n}{x - 1} > N[/itex]
[itex]\frac{1}{x - 1} > \frac{N}{n}[/itex]
[itex]x - 1 < \frac{n}{N}[/itex]
So choose [itex]\delta = \frac{n}{N}[/itex]
and [itex]0 < |x - 1| < \delta[/itex] holds
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My first question is of course, is this proof correct. Secondly, if it's not, my other question is, is choosing an n as I've done above, a legal operation? Could it be changed to make it a legal operation?
Thanks,
- Rob