Hard Limit to compute.

[Learningforcomp]

Hello.
During my practice of limits using L'Hopital's rule I managed to solve 9/10 problems But this one I cant figure it out , Can you help me?
The problem is :

benkei.png (10.39 KiB) Viewed 767 times

and the solution is =4/9.
What I did:
I noticed the indetermination of the type 0/0 and applied l'hopitals's rule, then I got to an expression of

flowey.png (2.6 KiB) Viewed 767 times

But I don't know what to do now I used the limit propertie to remove the 4 at the numerator out of the limit but When I try to plug in the - 1 It doesn't give me a result (1/5) that would give the solution when multiplied by the 4.
Could You please help me?
Your time is deeply appreciated.
Thank You

[nathi123]

A=[tex]\lim_{x \to -1}\frac{\sqrt[3]{1+2x}+1}{x+\sqrt{x+2}}=\lim_{x \to -1}\frac{(1+2x+1)(x-\sqrt{x+2})}{(\sqrt[3]{(1+2x)^{2}}-\sqrt[3]{1+2x}+1)(x^{2}-2-x)}=\lim_{x \to -1}\frac{2(x+1)(x-\sqrt{x+2})}{3(x+1)(x-2)}=\frac{2(1+\sqrt{3})}{9}[/tex].
because [tex](\sqrt[3]{1+2x}+1)(\sqrt[3]{(1+2x)^{2}}-\sqrt[3]{1+2x}+1)=1+2x+1=2(x+1) ;[/tex]
[tex](x+\sqrt{2+x})(x-\sqrt{2+x})=x^{2}-(2+x)=x^{2}-x-2=(x+1)(x-2)[/tex].