CTMC finding limiting distribution

Question

This problem is regarding Continuous time Markov Chains.

I have a rate matrix Q = $ \begin{bmatrix} -7/12 & 1/3 & 1/4 & 0 & 0 \\ 1/2 & -3/4 & 0 & 1/4 & 0\\ 1/2 & 0 & -5/6 & 0 & 1/3\\ 0 & 0 & 1/2 & -1/2 & 0\\ 0 & 1/2 & 0 & 0 & -1/2\\ \end{bmatrix} $

and initial distribution $\pi^{(0)} $ = (1 0 0 0 0)

I am tasked to find the limiting distribution and I am confused which method to use:

(1): $\pi^{(0)} \lim_{t\to 0} P_t$, where $P_t$ = $\sum^\infty_{k=0}\frac{(tQ)^k}{k!}$

I tried implementing this method into MATLAB using the expm function for the matrix exponential caluclation, but my MATLAB took too long to calculate, which made me think that this would be a wrong method. My code is as shown below:

syms t k 

Q = [-7/12, 1/3, 1/4, 0, 0;
     1/2, -3/4, 0, 1/4, 0;
     1/2, 0, -5/6, 0, 1/3;
     0, 0, 1/2, -1/2, 0;
     0, 1/2, 0, 0, -1/2];

initial_dist = [1, 0, 0, 0, 0];

Pt = symsum(expm(t*Q)/factorial(k),k,0,Inf);

eqn = limit(Pt, t, Inf);

%Limiting Distribution
LD = initial_dist * eqn

The second method I tried is this:

(2): $\vec\pi Q$= $\vec 0$

To my understanding, this finds the stationary distribution and cannot be assumed as the limiting distribution.

Can someone help me with this? Thank you.