btw L'Hopital rule is flawed i hope your not using that to determine if Mathematica is correct

your comparing points on two function lines, but l'hopital tries a trick using one line, the data isn't always there: L'Hopital's rules is plain wrong - wrong but easy!

The L'Hopital rule that I know is very correct. Perhaps you have another rule in mind.

There are different counterexamples to L'Hopital's rule, e.g. in:

"Counterexamples to L'Hopital's rule"

Bingo! Finally, I solved it after clarifying that Mathematica has two different cube roots for Power[ ] function and Surd[ ] function. In the above example, It must be used Surd[ ] function. So:

In[1]:= Limit[(Surd[15 + 2 x, 3] + 1)/(Surd[9 + x, 3] + x + 7), 
 x -> -8]

Out[1]= 1/2

Another strange result for the Limit function:

Limit[(Power[15 + 2 x, (3)^-1] + 1)/(Power[9 + x, (3)^-1] + x + 7), 
 x -> -8]

How to understand it?

Another limit, the same difficulty...

In the following example I intentionally gives the result at the beginning because it's not difficult to solve the problem analytically.

So, I tried to verify In Mathematica the result of this example

The Mathematica gives

In[1]:= Limit[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(Abs[
\*SuperscriptBox[\(E\), 
FractionBox[\(2.  \[Pi]\ I\ k\), \(n\)]] - 
\*SuperscriptBox[\(E\), 
FractionBox[\(2  \[Pi]\ I\ \((k - 1)\)\), \(n\)]]]\)\), 
 n -> \[Infinity]]

Out[1]= Limit[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(Abs[
\*SuperscriptBox[\(E\), 
FractionBox[\(\((0.`  + 6.283185307179586`\ I)\)\ k\), \(n\)]] - 
\*SuperscriptBox[\(E\), 
FractionBox[\(2\ I\ \((\(-1\) + k)\)\ \[Pi]\), \(n\)]]]\)\), 
 n -> \[Infinity]]

It's interesting that when I tried to compute simply the numerical evaluations of the expression under the limit, I obtained

In[2]:= Table[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(Abs[
\*SuperscriptBox[\(E\), 
FractionBox[\(2.  \[Pi]\ I\ k\), \(n\)]] - 
\*SuperscriptBox[\(E\), 
FractionBox[\(2  \[Pi]\ I\ \((k - 1)\)\), \(n\)]]]\)\), {n, 10000, 
10010}]

Out[2]= {6.28319, 6.28319, 6.28319, 6.28319, 6.28319, 6.28319, \
6.28319, 6.28319, 6.28319, 6.28319, 6.28319}

In[3]:= Table[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\(Abs[
\*SuperscriptBox[\(E\), 
FractionBox[\(2.  \[Pi]\ I\ k\), \(n\)]] - 
\*SuperscriptBox[\(E\), 
FractionBox[\(2  \[Pi]\ I\ \((k - 1)\)\), \(n\)]]]\)\), {n, 100000, 
100010}]

Out[3]= {6.28319, 6.28319, 6.28319, 6.28319, 6.28319, 6.28319, \
6.28319, 6.28319, 6.28319, 6.28319, 6.28319}

These results are approximately equal to $2\pi$. Why didn't work the function Limit[]?

Thank you, Mariusz. Is this a bug? Must we report it to WRI?

Agree!

Thank you, Gianluca, for a simple and clear answer to the question!

By default Limit takes the limit from the right. The option Direction is there for us to choose the direction: -1 is for the limit from the right (in the negative direction) and 1 for the limit from the left (in the positive direction):

Limit[(Log[1 - x] - Sin[x])/(1 - Cos[x]^2), x -> 0, Direction -> 1]
Limit[(Log[1 - x] - Sin[x])/(1 - Cos[x]^2), x -> 0, Direction -> -1]

It's NOT a error, because from Mathematica Documentation Center: