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 | High School Teacher | (Level 3) Associate Educator

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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!


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