If I remember correctly, the last time I wrote the article was two months ago. To refresh my mind, I will write this for the first time in a while.

This article may contain many grammatical errors and misspellings of words. If those things bother you, please correct the errors yourself and let me know in the comments, if you wish. I am confident that I often make mistakes even in Japanese. How cute I am.

First, I ask you a question. What is the derivative of $${x^2}$$.

That's right, the answer is $${2x}$$.

I suspect that the who be able to answer this correctly, perhaps, derived $${2x}$$ in a second. At least, they did not go back to the definition of derivative,

$$
\frac{\mathrm{d}}{\mathrm{d}x}x^{2} =\underset{h\to 0}{\lim}\frac{(x+h)^{2}-x^{2}}{h}.
$$

If someone bothers to consider as

“I must search a real number $${\alpha}$$ such that for any small positive number $${\epsilon}$$, there is an appropriate real number $${\delta}$$ satisfying

$$
0<|h|< \delta \Rightarrow \left|\frac{(x+h)^{2}-x^{2}}{h}
- \alpha\right|<\epsilon.
$$

So, I fix $${\epsilon>0}$$ arbitrarily. Then …”,

you are rather lovable silly than smart.

We can not spend time on every single derivative of $${x^2}$$, so everyone remembers the answer. Particularly important here is that being able to answer $${2x}$$ is irrelevant to knowing why they got that answer. This is similar to being able to drive a car even though you do not know how it works. If it works without any problems, that is all that matters.

Then, what about the following equation?

$$
\underset{n\to\infty}{\lim}\int\mathrm{d}xf_{n}(x) = \int\mathrm{d}x\underset{n\to\infty}{\lim}f_{n}(x).
$$

In this case, things are a little different. Since it is not obvious whether the integral and limit are possible to be interchanged, it should been examined how properties $${f_{n}}$$ has. However, if you were an engineer, you would believe that $${f_{n}}$$ has good enough properties and would not hesitate to swap the order. On the other hand, mathematicians and mathematical physicists will confirm that $${f_{n}}$$ converges almost everywhere to a certain function and is bounded by an integrable function. In other words, the former is more speedy and the latter is more careful in their calculations.

Something similar to the relationship between the engineers and mathematicians exists among English users. I think so at least.

Native English speakers speak English extremely fluently. They forget that "English" is a tool for conversation and use it like a part of their body. Most of them, however, do not care about grammatical rules and would never be able to explain them in the first place. In comparison, I am armed with a poor stick of "English," and I am mumbling something like incantations "articles... tenses... phase... aspect... the modals... marginal verbs... the catenatives... compound verbs... morphology..." and using it cautiously and in trepidation.

That is, I am mathematician-like in my use of the English language.

So, the first step in approaching native speakers is to not worry about every single detail, but the latent mathematician in me gets in the way. Since I keep coming up with these seemingly cool excuses, my English will never improve.

If I joke around too much, I might not really improve, so I would like to end here. Thanks for reading.