L'Hospital's Rule for a fraction where the numerator clearly grows faster than the denominator

Question

According to L'Hospital's Rule, the limit as x approaches infinity of $\frac{8x+5}{6x}$ is simply the derivative of the numerator over the derivative of the denominator is simply $8/6$. I don't understand why it is $8/6$. $8x+5$ clearly grows faster than $6x$, so shouldn't the limit be infinity?

Answer 1

1. ‘Grows faster’ doesn't means ‘grows infinitely faster’; i.e. if you know some asymptotic analysis, it doesn't mean, here, that $6x=o(8xx+5$. Actually, $6x=O(8x+5$.
2. Stop thinking that L'Hospital's rule is the ultimate way to limits: it is actually a dangerous rule, as it requires some hypotheses that most people never check. Anyway, when it works, it is equivalent to Taylor's formula at order $1$.
3. It is a high-school theorem that the limit at infinity of a rational function is just the limit of the ratio of the leading terms of its numerator and denominator, so that all you have to say here is this: $$\lim_{x\to\infty}\frac{8x+5}{6x}=\lim _{x\to\infty}\frac{8\not x}{6\not x}=\frac 43.$$

Answer 2

Without Using L'hopital $$ \lim_{x\to\infty}\frac{8x+5}{6x}= \lim_{x\to\infty}\frac{8+5x^{-1}}{6}=\frac{8}{6} $$ as $\lim_{x\to\infty}x^{-1}=0$.