1. **State the problem:** We need to find the length of side $x$ in a right triangle where the angle is $68^\circ$, the opposite side is $12.6$, and the adjacent side is $x$. The tangent ratio is given by $$\tan 68^\circ = \frac{12.6}{x}.$$\n\n2. **Recall the tangent formula:** For a right triangle, $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}.$$ Here, $\theta = 68^\circ$, opposite side = $12.6$, and adjacent side = $x$.\n\n3. **Set up the equation:** $$\tan 68^\circ = \frac{12.6}{x}.$$\n\n4. **Solve for $x$: $$x = \frac{12.6}{\tan 68^\circ}.$$\n\n5. **Calculate $\tan 68^\circ$: Using a calculator, $$\tan 68^\circ \approx 2.4751.$$\n\n6. **Substitute and simplify:** $$x = \frac{12.6}{2.4751}.$$\n\n7. **Show cancellation:** $$x = \frac{12.6}{\cancel{2.4751}} \times \frac{1}{\cancel{2.4751}} = 5.09.$$\n\n8. **Final answer:** The length of side $x$ is approximately $$\boxed{5.09}.$$