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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).

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After using it a few times, you will recognize this as one of the many "tricks of the trade," and will be able to use it comfortably. And after a bit longer, you will be able to invent similar devices. – André Nicolas Jun 18 '12 at 6:08
Regarding your concern I understand this solution once I read the answer, but I commonly don't see the path or logic to arriving at it, perhaps my comments in this 26 Sept. 2011 AP-calculus post (about uniformly rescaling collections of numbers) could be helpful. – Dave L. Renfro Jun 18 '12 at 15:14
up vote 4 down vote accepted

What you're looking for is a heuristic for solving limits, and in the future, other problems in math, and calculus. I am not aware of attempts made to actually write down such heuristics, but I can try to offer some general advice:

  1. Don't be afraid to "play" with the problem, rather than concentrating on "solving" it. This includes inserting real numbers to get a "feel" for the problem, rewriting it in other terms etc.
  2. Practice, Practice, Practice - solving a large amount of problems, with occasionally peeking at the solutions, is perfectly normal, and is really the best way of picking up the relevant heuristics on your own. Unless you've attempted to solve tens or hundreds of problems, I would definitely not say your mindset is "lacking".

As to some heuristics for calculating limits:

  1. Factoring - as in your example above, factoring leads to clearly seeing the "parts" of the problem, some finite and some infinite. This is especially usefull for added fractions in the limit.
  2. Division by the largest power - This can help solve problems of the form: $$\lim_{x\rightarrow y}\frac{x^3+x}{x^4+x+1}$$
  3. inverse - switch $x$ into $1/x$, and the limit accordingly.
  4. replace - Sometimes switching some function of $x$ with a different variable (say $y$) helps "see" what the limit really is, e.g: $$\lim_{x\rightarrow \infty}{(1+b/x)^x}, y = x/b$$

There are many more, but with enough practice, you'll find the ones you like best.

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The heuristics really nail how limits are computed in calculus courses; I wish my professor told me them when I took my first calculus course... – Alex Nelson Jun 18 '12 at 8:08

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