This question is an extension of Endogenous Growth: Balanced Growth Path with CRRA Utility however, this question asks about a specific concept used in that question, and I think it would be helpful to have a question dedicated to this concept.
We will use the model in this question:
$\textbf{Model:}$ $$K_t=\frac{1}{n}\sum_{t=1}^nk_t$$ In this model, $k_t$ is chosen by agents, and $K_t=\bar{k}_t$ (the average of all $k_t$).
Now, agents want to dynamically maximize utility (under certain constraints) and they have CRRA (constant relative risk aversion) utility, so the maximization looks like: $$\sum_{t=0}^\infty\beta^t\bigg(\frac{c_t^{1-\gamma}}{1-\gamma}\bigg)$$ $$s.t.\;Y_t=k_t^\alpha(E_tL)^{1-\alpha}$$ $$c_t+i_t=Y_t$$ $$k_{t+1}=(1-\delta)k_t+i_t$$ $$c_t,i_t\geq0$$
$E_tL$ is effective labor and the rest of the variables are typical (I can give their definitions if requested).
One last addition to the model is that there are two equilibrium constraints: $$E_t=\frac{K_t}{L}$$ $$k_t=K_t$$
In the answers section, I was instructed to use transversality conditions and inada conditions in order to show that a balanced growth path is optimal. Below is my derivation of the transversality condition:
First, we must derive the finite horizon transversality condition. We do this by solving: $$\underset{\{k_t\}_{t=0}^{T+1}}{max}\;\sum_{t=0}^{T}\beta^tU(\frac{}{})$$ $$s.t.\;k_{T+1}\geq0$$ With our model that means: $$\underset{\{k_t\}_{t=0}^{T+1}}{max}\;\sum_{t=0}^{T}\beta^t\bigg(\frac{((1+1-\delta)k_t-k_{t+1})^{1-\gamma}}{1-\gamma}\bigg)$$ $$s.t.\;k_{T+1}\geq0$$
Our Lagrangian is: $$\ell(\frac{}{})=\sum_{t=0}^{T}\beta^t\bigg(\frac{((1+1-\delta)k_t-k_{t+1})^{1-\gamma}}{1-\gamma}\bigg)+\lambda k_{T+1}$$
In order to get the transversality condition, we differentiate with respect to $k_{T+1}$ and use the constraint.
FOC: $$\frac{\beta^T}{((1+(1-\delta))k_T-k_{T+1})^\gamma}=\lambda \qquad (1)$$ $$\lambda k_{T+1}=0 \qquad (2)$$ $$\lambda , k_{T+1}\geq 0 \qquad (3)$$ Solving this system, we get $\textbf{the finite horizon transversality condition is:}$ $$\bigg(\frac{\beta^T}{((1+(1-\delta))k_T-k_{T+1})^\gamma}\bigg)k_{T+1}=0$$ Now, in order to get the infinite horizon transversality condition, we maximize the discounted utility function again, but we set the limit of finite horizon transversality condition as $T\rightarrow \infty$ equal to 0 as the constraint.
This means that we have to solve: $$\underset{\{k_t\}_{t=0}^\infty}{max}\;\sum_{t=0}^\infty\beta^t\bigg(\frac{((1+(1-\delta))k_t-k_{t+1})^{1-\gamma}}{1-\gamma}\bigg)$$ $$s.t.\; \underset{T\rightarrow\infty}{lim}\;\bigg[\bigg(\frac{\beta^T}{((1+(1-\delta))k_T-k_{T+1})^\gamma}\bigg)k_{T+1}\bigg]=0$$
This is where my question comes. How do we solve this? Can we use a lagrangian? If so, how do we take first order conditions in the presence of the limit? Any help would be greatly appreciated!