When we say as a variable n tends to infinity, we mean as n gets very very large. For example. If we look at 1/n as n tend to infinity, then n gets very large and 1/n goes to zero.
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As x tends towards 0 (from >0), log(x) tend to - infinity. As x tends to + infinity so does log (x), though at a much slower rate.
There is no answer to this question. The answer tends to infinity.
I believe the maximum would be two - one when the independent variable tends toward minus infinity, and one when it tends toward plus infinity. Unbounded functions can have lots of asymptotes; for example the periodic tangent function.
I think the phrase is "a line that tends towards infinity", but I'm not sure.
(xn) is Cauchy when abs(xn-xm) tends to 0 as m,n tend to infinity.