What is the limit $\lim_{x \to 0^+}\log(x) + \frac{c}{x}$ where $c > 0$?

Question

$\frac{c}{x}$ approaches positive infinity much faster than $\log(x)$, so I suspect the limit should be positive infinity. But how to derive it from the rules of limit calculation please?

Answer 1

$1/x=m\to\infty$;$$\lim_{m\to\infty}cm-\log m=\lim_{m\to\infty}m\left(c-\frac{\log m}m\right)$$Use standard limits to continue.

Answer 2

$e^{\log\, x+\frac c x}=\frac {e^{c/x}} {1/x}$. Apply L'Hopital's Rule to see that the limit of this expression is $\infty$. Hence the given limit is also $\infty$.