Understand the growth of $(\ln x)^N$ where N is a natural number

Question

Let $f$ and $g$ be functions defined on $(1,∞)$ and assume they are always positive. We say that $f$ grows much slower than $g$ if $$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$$ For any $N \in \Bbb{N}$ and $\delta > 0$, show that $(\ln x)^N$ grows much slower than $x^\delta$. You can think of N as large and $\delta$ as small.

The question I have is that I don't understand how to approach this problem.I already have trouble completely understanding ln and log functions. This question is based on our lessons on application of derivatives.

Answer 1

Consider the sequence defined by $x_i=2^i$, and define also $y_i=\frac{(\ln x_i)^N}{x_i^\delta}.$ We’ll prove $$\lim_{i\to\infty}y_i=0,$$ thus proving what we want.

We have $$y_i=\frac{\left(\ln x_i\right)^N}{x_i^\delta}=\frac{\left(i \ln 2\right)^N}{2^{\delta i}}.$$ This means that the ratio between two consecutive $y_i$ terms is $$\frac{y_{i+1}}{y_i}=\frac{\left(\frac{i+1}i\right)^N}{2^\delta}.$$ As $i\to\infty$, this ratio tends to $\frac1{2^\delta}<1$. Therefore, $y_i$ tends to zero, as we wanted.

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In reality, this is slightly incomplete, as we didn’t prove that the limit exists, we just found its value given that it did. However, the argument can easily be adapted; we can prove that $\frac{\left(\ln 2^x\right)^N}{2^{x\delta}}$ is decreasing for large enough $x$ in almost the same manner. With this, the limit is formally established. $\blacksquare$

Answer 2

Some approaches you can take to this problem:

So the quotient limit becomes

$$\lim_{x\to \infty}\left\{\frac{\ln(x)^N}{x^\delta}\right\}$$

By definition of exponential function of basis b: $b^t = \exp(t\cdot \ln(b))$

$$\lim_{x\to \infty}\left\{\frac{\exp(N\cdot \ln(\ln(x)))}{\exp(\ln(x)\delta)}\right\} = \lim_{x\to \infty}\left\{\exp(N\cdot \ln(\ln(x))-\ln(x)\delta)\right\} $$

Can you finish from here?

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Maybe it gets easier to write like this :

$$\lim_{x\to \infty}\left\{\ln(x)^Nx^{-\delta}\right\}$$

We can differentiate this, we get:

$$-x^{-\delta-1}\log(x)^{N-1}(x)(\delta\log(x)-N)$$

This, we should be able to show goes to 0 with help of some standard limits. So our function approaches a constant value. Can we show this constant value can't be anything else than 0 then we are done.