Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq 1,\quad X_0=0 $$
Find $(X_t)_{0 \leq t \leq 1}$ and verify that
$$ L^2\text{-}\lim_{t \to 1}X_t=0$$ $X$ is called Brownian Bridge.
I will give you an answer on the general brownian bridge case.
Consider the SDE
$$dX_t =\frac{b-X_t}{1-t}dt+dW_t,\ X_0=a$$
for $t\in [0,1]$ with $a,b \in \mathbb R $.
An approach to solve this SDE can be obtained by the constant variation method. Indeed, consider the following ODE
$$ x'(t)= \frac{b-x(t)}{(1-t)} + f(t), \ x(0)=a$$ for $t\in [0,1]$. We can easily obtain by constant variation method the solution of this equation which is given by
$$x(t) = a(1-t) +bt +(1-t)\int_0^t \frac {f(s)}{1-s}ds$$
At this point, we would like to simply apply this result with $f(t)=dW_t /ds$, which is formally not possible. But let allow us to consider this formula anyway and then prove that
$$X_t = a(1-t) +bt +(1-t)\int_0^t \frac {1}{1-s}dW_s\tag{1}$$
is a solution to our SDE. By effect, a stochastic integration by parts give us that
$$X_t = a(1-t) +bt +W_t-(1-t)\int_0^t \frac {W_s}{(1-s)^2}ds.$$
Then, if we consider $Y_t = X_t-W_t$, we have $$ dY_t = \frac{b-Y_t}{1-t}dt - \frac{W_t}{1-t}dt$$ so we can easely conclude the wanted result.
Now, knowing that the brownian bridge have the form given by $(1)$ we are able to calculate it's $L^2-$limit. Note that $X$ defined by $(1)$ is a gaussian process with mean $a(1-t)+bt$. Using Ito's isometrie, we can straight forward calculate $$ \mathbb E \{ X_t^2\}=\left[a(1-t)+bt\right]^2 + (1-t)^2\int_0^t \frac{1}{(1-s)^2}ds=\left[a(1-t)+bt\right]^2 +t(1-t)\underset{t\rightarrow 1}{\longrightarrow }b^2$$
wich show us that $L^2-\lim_{t \rightarrow 1} X_t =b$ (with $b=0$ in your case).
Actually this limit holds even $\mathbb P-\text{almost ever}$.