A graph shows the limit to be 1/2.
https://www.desmos.com/calculator/qrf6ay47tw
Since the value of the function is the indeterminate form 0/0, L'Hôpital's rule applies. The ratio of derivatives of numerator and denominator is
.. x/

Evaluated at x=1, this is
.. 1/

= 1/2
This is a matter of plugging in. Specifically 7 for L.
T = 2πsqrt(7/32)
T = (6.28319...)(0.46771...)
T is about 2.93871 seconds
Let's verify by substituting 1 to x.
2 - 8x = -6
2 - 8(1) = -6
2 - 8 = -6
-6 = -6
It is equal, therefore x = 1 is the solution.