Question

What is the limit of `((1)/(x))-((1)/(e^(x)-1))` as x approaches infinity?

Answer 1

If two limits added together individually approach 0, the whole thing approaches 0.

Use the property that limits distribute over addition and subtraction.

`=> lim_(x->oo) 1/x - lim_(x->oo) 1/(e^x - 1)`

The first limit is trivial; `1/"large" ~~ 0`. The second one asks you to know that `e^x` increases as `x` increases. Hence, as `x->oo`, `e^x -> oo`.

`=> color(blue)(lim_(x->oo) 1/x - 1/(e^x - 1))`

`= 1/oo - 1/(oo - cancel(1)^"small")`

`= 0 - 0 = color(blue)(0)`