Find the limit, if it exists. (If an answer does not exist, enter DNE.)`lim_(x->0^+) (8/x)/(8/(lxl))`  

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txmedteach eNotes educator| Certified Educator

To determine this answer, we can actually start by considering what it means for this limit to approach zero from the right.

If we're approaching zero from the right, then x will always be positive!

Our limit becomes:

`lim_(x->0^+) (8/x)/(8/x)`

Well, this limit is actually fairly easy to evaluate at this point. Consider what it means to divide by a fraction:

`lim_(x->0^+) 8/x * x/8`

Clearly the 8's cancel leaving us with:

`lim_(x->0^+) x/x`

Well, we could just wave our hands and say "x will never be 0, so we can just cancel the terms out to get 1."

However, let's just be a little more formal and go ahead and recognize that our limit as it is becomes `0/0`. This indeterminant form dictates that we can equate our limit to the derivatives of the top and bottom (by L'Hopital's Rule):

`lim_(x->0^+) x/x = lim_(x->0^+) 1/1 = 1`

And there you have it. Using L'Hopital's Rule we can easily find the final value of our limit.

Just to sum things up:

`lim_(x->0^+) (8/x)/(8/(|x|)) = 1`

I hope that helps!