Need help calculating limit

Hi everyone,

I found this limit in a research paper, and I'm simply not able to get to the same result.

I hope some of you can help me a little bit.

Hypothesis from the paper:

$\displaystyle \lim_{\beta \rightarrow 0} \frac{1}{\alpha} (C^{\beta}+\theta L^{Beta})^{\frac{\alpha}{\beta}} = \frac{1}{\alpha}(C^{\frac{1}{1+\theta}}+\theta L^{\frac{\theta}{1 + \theta}})^{\alpha} $

My idea so far, was to take the logarithm of the problem, and then use some de l'Hôpital:

$\displaystyle \lim_{\beta \rightarrow 0} \frac{1}{\alpha} (C^{\beta}+\theta L^{Beta})^{\frac{\alpha}{\beta}} = e^{\left( \lim_{\beta \rightarrow 0} log\left(\frac{1}{\alpha} (C^{\beta}+\theta L^{Beta})^{\frac{\alpha}{\beta}}\right) \right)}$

$\displaystyle \lim_{\beta \rightarrow 0} log\left(\frac{1}{\alpha} (C^{\beta}+\theta L^{Beta})^{\frac{\alpha}{\beta}}\right) = \lim_{\beta \rightarrow 0} \frac{log(1+\theta)}{\beta} + \lim_{\beta \rightarrow 0} \frac{(\frac{1}{(1+\theta)}K^{\beta} + \frac{\theta}{(1+\theta)} L^{beta})}{\beta}$

Using de l'Hôpital on the second term yields:

$\displaystyle = \lim_{\beta \rightarrow 0} \frac{log(1+\theta)}{\beta} + \frac{1}{1+\theta} log(K) + \frac{\theta}{1+\theta}log(L)$

As you can see, the first term is divergent, which should mean that the limit doesn't exist.

What do you think ?

Is the paper wrong? Am I wrong ?

Thanks a lot in advance

El Botas