Transversality Condition in neoclassical growth model In the neo-classical growth model there is the following transversality condition: 

$$\lim_{t\rightarrow\infty}\beta^{t}u'(c_{t})k_{t+1}= 0,$$ 
where $k_{t+1}$ is the capital at period $t$. 

My questions are: 


How we derive this condition? 
Why do we require this, if we want to rule out paths with no debt accumulation?
Why are the Lagrange multipliers $\beta^{t}u'(c_{t}) = \beta^{t}\lambda_{t}$
the present discounted value of the capital?
The transversality condition may be more easily understood if we start from a problem with finite horizon.

In the standard version, our objective is to 
$$ 
\max_{\{c_t,k_{t+1}\}_{t=0}^T} \sum_{t=0}^T\beta^t u(c_t)
$$
subject to 
$$
\begin{aligned}
f(k_t)-c_t-k_{t+1}&amp;\ge0,\quad t=0,\dots,T &amp;&amp;\text{(resource/budget constraint)}\\
c_t,k_{t+1}&amp;\ge0,\quad t=0,\dots,T &amp;&amp;\text{(non-negativity constraint)}
\end{aligned}
$$
with $k_0$ given. The associated Lagrangian (with multipliers $\lambda_t$, $\mu_t$, and $\omega_t$) is 
$$
\max_{\{c_t,k_{t+1},\lambda_t,\mu_t,\omega_t\}_{t=0}^T}
\sum_{t=0}^T \beta^tu(c_t)+\lambda_t(f(k_t)-c_t-k_{t+1})+\mu_tc_t+\omega_tk_{t+1}
$$
The FOCs are 
$$
\begin{align}
c_t:&amp;&amp; \beta^tu'(c_t)-\lambda_t+\mu_t&amp;=0,\quad t=0,\dots,T \\
k_{t+1}:&amp;&amp; -\lambda_t+\lambda_{t+1}f'(k_{t+1})+\omega_t&amp;=0,\quad t=0,\dots,T-1 \\
k_{T+1}:&amp;&amp; -\lambda_T+\omega_T&amp;=0,\quad T+1 \tag{1}
\end{align}
$$ 
with the Kuhn-Tucker complementary slackness conditions: for $t=0,\dots,T$,
$$
\begin{align}
\lambda_t(f(k_t)-c_t-k_{t+1})&amp;=0 &amp; \lambda_t&amp;\ge0 \\
\mu_tc_t&amp;=0 &amp; \mu_t&amp;\ge0\\
\omega_tk_{t+1}&amp;=0&amp;\omega_t&amp;\ge0\tag{2}
\end{align}
$$
Since resource constraint must be binding in all periods, i.e. $\lambda_t&gt;0$ for all $t$, it follows that at the last period $T$, $\omega_T=\lambda_T&gt;0$, which in turn implies $k_{T+1}=0$.

Usually we assume $c_t&gt;0$ for all $t$ (the Inada condition), and this implies $\mu_t=0$ for all $t$. So the consumption FOC becomes
$$
\beta^tu'(c_t)=\lambda_t \tag{3}
$$

Looking at conditions $(1)$ $(2)$ and $(3)$ in the last period $T$, we get
$$\beta^Tu'(c_T)k_{T+1}=0$$
Extending this to the infinite horizon, we get the transversality condition
$$\lim_{T\to\infty}\beta^Tu'(c_T)k_{T+1}=0$$

The intuition of the transversality condition is partly that "there is no savings in the last period". But as there is no "last period" in an infinite horizon environment, we take the limit as time goes to infinity.