a. Assuming that limn →∞(1 / n^f)=0 if c is any positive constant, show that limn →∞ (lnn)/(n^f)

step 1 Hello. So here in part A, we have the limit as n goes to infinity of the natural log of n divide

A. Assuming that limn →∞(1 / n^f)=0 if c is any positive...

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{f}\right)=0$ if $c$ is any positive constant, show that $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{f}}=0$$ if $c$ is any positive constant. b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{\circ}\right)=0$ if $c$ is any positive constant. (Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to ensure that $\left|1 / n^{r}-0\right|<\epsilon$ if $n>N ? )$

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{f}\right)=0$ if $c$ is any positive constant, show that
$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{f}}=0$$
if $c$ is any positive constant.
b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{\circ}\right)=0$ if $c$ is any positive constant.
(Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to
ensure that $\left|1 / n^{r}-0\right|<\epsilon$ if $n>N ? )$

Key Concepts

Limit of Sequences

This concept involves determining the behavior of a sequence as its index approaches infinity. In the context of these problems, it is fundamental to understand that if the denominator of a fraction grows much faster than the numerator as the index increases, the entire fraction converges to zero. This includes understanding that sequences like 1/n^c, when c is a positive constant, tend to zero as n becomes arbitrarily large.

Growth Rates of Functions

In the study of limits, comparing the growth rates of different functions is critical. For example, logarithmic functions like ln(n) grow much more slowly than power functions such as n^c when c > 0. This difference in growth rates is essential for proving that the ratio ln(n)/n^c approaches zero because the denominator increases significantly faster than the numerator. This concept is a key part of understanding how different types of functions behave as their input values become large.

Epsilon-N Definition of Limits

The epsilon-N definition provides a rigorous framework for proving the limits of sequences. It states that for every ? > 0, there exists an N such that for all n > N, the absolute difference between the sequence term and the limit is less than ?. This method is used to formally prove that sequences like 1/n^c converge to zero by identifying how large n must be to ensure that the sequence terms fall within any arbitrarily small tolerance of the limit.

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