limit - show that in infinity function grow faster that other function

Question

given two function:

1. $x\:\le\:b\cdot \log\left(x\right)\:$

2. $x\:\le\:x\cdot \:\log\left(x\right)\:$

i want to show that $b\cdot \:\log\left(x\right)\:\le \:x\cdot \log\left(x\right)\:$

I made a graph of these 2 functions, you can clearly see that this is true, at infinity, this equation (2) is always above this equation (1)

I was thinking of showing the limit at infinity: $\lim _{x\to \infty }\left(\left(b\cdot \:\log\left(x\right)\:\right)/\left(\:x\cdot \:\log\left(x\right)\:\right)\right) = 0$

I'm not sure if this is a correct way to do it, am I right?

Answer 1

This is a correct way, which basically states that $b\cdot ln(x)$ gets infinitely smaller than $x\cdot ln(x)$ when $x$ grows infinitely large. Or more informally, the more you blow $x$ up, the value of $b\cdot ln(x)$ will be more negligible than $x\cdot \ln(x)$.

This can be easily shown using the way you stated: $$\lim_{x\to\infty} \frac{b\cdot \ln(x)}{x\cdot \ln(x)} $$ $$=\lim_{x\to\infty} \frac{b}{x} $$ $$ = 0$$ since $b$ is just a constant.