Use formal definitions to prove the limit statement:  lim x->0 of 1/|x|=`oo`  I know I have to use the epsilon delta inequality, but I don't know how to go about doing that

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Roughly speaking, we need to show that we can make the quantity `1/|x|` as large as we want by taking `x` sufficiently close to 0. More precisely, we want to show that for any given `M>0` , there exists some `delta>0` such that

`1/|x|>M` for all `|x|<delta`. Note that  `1/|x|>M` is equivalent to `|x|<1/|M|` , which suggests that we choose `delta=1/M` . Then the proof would just go something like:

Let `M>0` be given. Set `delta=1/M` . Then if `|x|<delta=1/M` , we see that `1/|x|>M` , so `lim_(x->0) 1/|x|=oo`.

One more remark. The page I link to uses `epsilon`, where I've used `M` instead. It makes no difference of course, but since ` ` we usually think of `epsilon` as a "small" number, here we intuitively think of `M` as a large number. That's the only reason I used `M`.

Sources:

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