The limit ${\displaystyle \lim_{x\rightarrow 0}\frac{(60+2x^4)(2-x^4-2\cos{x^2})+5x^8}{5x^8(2-x^4-2\cos{x^2})}}$ is a candidate for evaluation by L'Hôpital's rule -- it falls into the $\frac{0}{0}$ indeterminacy case. And with extreme patience (or a CAS) this limit can indeed be successfully evaluated by iterative application of L'Hôpital's rule. However, as evaluating the Sage cell below shows, it takes sixteen applications of L'Hôpital's rule to escape from the $\frac{0}{0}$ form! You can uncomment the commented-out print statement to see just how horrendously messy the derivative computations become.
For those who insist on finding the limit without machine assistance, the theory behind Taylor Series (covered in Calc 2) will provide a much more efficient method of computation.