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When is it valid to deal with a term as a "negligible" one in a limit?

I am asking this question because I usually do not take limits very seriously, and I can do a lot of "illegal" moves just to evaluate and get back to the important thing.

This is a very broad question, that is why I'll just give two examples and ask about them instead.

Example $1$:

$$\lim_{n \to \infty} \frac{e^n + n^{2015} + 1}{e^\sqrt{n^2 + \sin(n) + \ln(n)}}$$

I would usually say: the numerator is asymptotic to $e^n$, and the denominator is as well, because $n^2 + \sin(n) + \ln(n)$ is asymptotic to $n^2$.

The limit eventually becomes $1$.

Had I done an invalid move?

Example $2.1$:

$$\lim_{n \to \infty} \frac{n+1}{n} \times |x|^{n} = \lim_{n \to \infty} |x|^{n}$$

Example $2.2$:

$$\lim_{n \to 0} \frac{n + 1}{n + 2} \times \frac{\ln(n)}{e^n} = \frac12 \lim_{n \to 0} \frac{\ln(n)}{e^n}$$

Was my writing in $2.1$ and $2.2$ valid?

Please elaborate, and if possible, warn me about some common misconceptions regarding these things.

Thanks a lot.

1 Answers1

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The reasoning of the first problem is dangerous. For example, $n^2+2n+1$ is "asymptotic" to $n^2$. But $\frac{e^n}{e^{\sqrt{n^2+2n+1}}}=e^{-1}$ for all $n$.

Reasoning of the type used in 2.1 and 2.2 is quite safe. If we have a product or sum, we can separate out "well-behaved" portions.

André Nicolas
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  • Very good example. So, how can one know whether or not eliminating a term, by means of the "asymptotic to" reasoning, is safe in a certain limit? –  May 24 '15 at 00:48
  • Experience helps. After a while one can, at least in not too subtle cases, tell when something is negligible. There are also general theorems. For example, if $\lim_{x\to a}f(x)=c\ne 0$, then $\lim_{x\to a}f(x)g(x)=c\lim_{x\to a}g(x)$. That sort of thing is behind the validity of your 2.1 and 2.2. – André Nicolas May 24 '15 at 01:01