limit of the ratio of two functions of log(x) as x goes to 0 from the positive-x side?

What assumptions, if any, are required such that the following limit is correct?

limx→0+alog(x)blog(x)=ablimx→0+alog(x)blog(x)=ab\lim_{x \rightarrow 0^+}{\frac{a \log(x)}{b \log(x)} } = \frac{a}{b}




What needed is b≠0b≠0b\ne 0.
– Ng Chung Tak
2 days ago


1 Answer


The only assumption needed is b≠0b \neq 0.

Since we are approaching 0 from the right, log(x)log(x) is defined for all x>0x>0, there is no problem there. Thus, the log(x)log(x) can be cancelled from bottom and top and we are left with ab\dfrac{a}{b}, which is only defined if b≠0b\neq 0.