Okay, let's see if we can do this without L'Hôpital's Rule.
Let $v=u+1$ and write the limit as:
$$L=\lim_{v\to1}\left(\frac{v^{\frac{1}{2}}-v^{\frac{1}{3}}}{v-1}\right)$$
Now, let's try to rewrite the expression so that it is not an indeterminate form:
$$\frac{v^{\frac{1}{2}}-v^{\frac{1}{3}}}{v-1}\cdot\frac{v^{\frac{1}{2}}+v^{\frac{2}{3}}}{v^{\frac{1}{2}}+v^{\frac{2}{3}}}=\frac{v^{\frac{5}{6}}\left(v^{\frac{1}{3}}-1\right)}{(v-1)\left(v^{\frac{1}{2}}+v^{\frac{2}{3}}\right)}$$
Next, let's observe that $v-1$ may be written as the difference of cubes and then factored:
$$v-1=\left(v^{\frac{1}{3}}\right)^3-1^3=\left(v^{\frac{1}{3}}-1\right)\left(v^{\frac{2}{3}}+v^{\frac{1}{3}}+1\right)$$
Hence, we have:
$$\frac{v^{\frac{1}{2}}-v^{\frac{1}{3}}}{v-1}=\frac{v^{\frac{5}{6}}}{\left(v^{\frac{2}{3}}+v^{\frac{1}{3}}+1\right)\left(v^{\frac{1}{2}}+v^{\frac{2}{3}}\right)}=\frac{v^{\frac{1}{3}}}{\left(v^{\frac{2}{3}}+v^{\frac{1}{3}}+1\right)\left(v^{\frac{1}{6}}+1\right)}$$
And thus, our limit is:
$$L=\lim_{v\to1}\left(\frac{v^{\frac{1}{3}}}{\left(v^{\frac{2}{3}}+v^{\frac{1}{3}}+1\right)\left(v^{\frac{1}{6}}+1\right)}\right)$$
Now we no longer have an indeterminate form, and we may evaluate the limit directly. :D
edit: I've moved this thread from our Analysis forum to our Calculus forum since that's a better fit for the thread topic. :D