Find the limit as x approaches 0+ of (lnx)/x using L'hospitals rule.
When I do this, I keep getting stuck at 1/0 when you plug back into the equation after doing l'hospital once.
Use L'hopitals rule to find the limit.
If you take the derivative of the top, and the derivative of the bottom
derivative of lnx = (1/x)
and the derivative of x = 1
You should be able to eliminate your problem with the 0.
So, why is ∞ not a valid answer?
Sometimes that is the limit.
lHospital does not guarantee that the limit is defined. It just allows one to make sure that an indeterminate form is avoided. 1/0 is not indeterminate -- it is infinite.
The only thing that bothers me is that
lnx/x -> -∞/0 = -∞
but 1/0 = +∞
when x = δ, a very small positive value.
any ideas on that?
Whoops, sorry about that. It does seem like you still wind up with 1/0.
Hmm... let's see when you graph it it looks like it's approaching -infinity so that's probably why you keep winding up with 1/0 since that will give you infinity technically.
Also, I don't believe there's any way to further simplify this problem so that you don't wind up with 1/0.
If you graph it you'll see what I'm talking about.
I am not sure L'Hopital's rule is relevant here because the limit of the top derivstive is -oo and of the bottom is 1 but if you try it with numbers and your calculator you will indeed get -oo
for example
ln .0001 = -9.21
-9.21/.0001 = -92103
To find the limit as x approaches 0+ of (lnx)/x using L'Hôpital's Rule, you need to apply the rule repeatedly until you no longer end up with an indeterminate form.
Here's a step-by-step breakdown:
Step 1: Determine the indeterminate form.
Evaluate the limit as x approaches 0 of (lnx)/x. Plugging in 0 would result in an indeterminate form of 0/0.
Step 2: Apply L'Hôpital's Rule once.
Take the derivative of both the numerator and the denominator: d/dx (lnx) = 1/x, d/dx (x) = 1.
Now evaluate the limit of the derivatives: lim(x->0) (1/x) / (1) = lim(x->0) 1/x.
However, you are still left with an indeterminate form of 1/0.
Step 3: Apply L'Hôpital's Rule again.
Differentiate the numerator and denominator once more: d/dx (1/x) = -1/x², d/dx (1) = 0.
Now calculate the limit of the new derivatives: lim(x->0) (-1/x²) / (0) = lim(x->0) (-1/x²).
Note that this is no longer an indeterminate form, as the denominator is now approaching infinity.
Step 4: Evaluate the actual limit.
At this point, you can directly compute the limit without encountering an indeterminate form: lim(x->0) (-1/x²) = -∞.
Therefore, the limit as x approaches 0+ of (lnx)/x using L'Hôpital's Rule is -∞.
If you were encountering difficulties after applying L'Hôpital's Rule once, it is possible that you made an error in differentiating the functions or simplifying the expression. Make sure to carefully apply the rule and double-check your calculations.