Recently I have been spending around 15-20 hours a week learning calculus, and I have to admit that the process, while I enjoy it, is seemingly slow going. I come back to limit questions, and I can observe that my algebraic skills have advanced, but my conceptual understanding of how to solve the questions feels to be stagnant. As an example, I was asked to find the limit of $\lim_{x\to -\infty}(x^4+x^5)$
When I saw this I was thrown off, and I peeked at the solution - which said to remove the largest power of $x$. $\lim_{x\to -\infty}x^5\left(\frac{1}{x}+1\right)$
From here the logic is that $\lim_{x\to -\infty}x^5=-∞$
and since, $\lim_{x\to -\infty}\left(\frac{1}{x}+1\right)=1$
$\lim_{x\to -\infty}x^5\left(\frac{1}{x}+1\right)=-\infty$
I understand this solution once I read the answer, but I commonly don't see the path or logic to arriving at it. And this makes me feel as though my mindset towards problem solving is lacking a footing in the concepts underlying limits.
I would like to introduce an analogy to try and better articulate what I am getting at. When I was first learning to play guitar my teacher asked me to hold my hand in a position which did not feel totally comfortable or natural, and had me fret notes with my pinky, which was weak in comparison to my other fingers. Fretting with the pinky was difficult and I didn't get why I should use it when my index and middle and ring finger where much stronger. But over time my pinky strengthened, as did my coordination, and had I not developed the use of my pinky, I would have had a major handicap in my playing.
Perhaps this is a poor analogy. But what I am trying to determine is if there is a better way to learn limits, derivatives, concepts in calculus, and even more generally topics in mathematics (taking into account that any method is going require countless hours of hard work and effort).