Does the infinite series 9 divided by 10 to the power of n starting from 1 converge to 1 proof

For limits, I would first classify the expression before applying any rule. If direct substitution gives a number, you are done. If it gives $0/0$ or $\infty/\infty$, then algebra, series expansion, or L'Hopital's rule may be useful. If it gives something like $0\cdot\infty$ or $\infty-\infty$, rewrite it into a quotient first. For example, near $0$ the expansion $$\sin x=x-\frac{x^3}{6}+O(x^5)$$ often gives the cleanest answer. It also explains why repeated L'Hopital works in some cases but can hide the structure of the expression.