Question

Use L'Hôpital's rule to solve these questions:

I'm going to put them in a second

Answer 1

\[\lim_{x \rightarrow 0} \frac{ x cosx - sinx }{ xsin ^{2}x }\]

Answer 2

this is the first question , I already know the answer , I want someone to explain , because in my notes " from my prof " he said that the answer is -1/3
and the final step is going to be:
\[\frac{ -1 }{ 2x sinx+3cosx }\]

and then we find the limit to get -1/3

but when I tried to solve it , I got :

\[\frac{ 1 }{ 2+sinx cosx }\]

and I got 1/3 as the answer.

Answer 3

the second questions :

also using the same rule to find :

\[\lim_{x \rightarrow 0+} xLnx\]

Answer 4

about the first question , the last step that I did is going to be :

\[\frac{ 1 }{ 2+\cos ^{^{2}}x } then find the limit\]