In taking the limit, we're considering [tex]x[/tex] as it gets arbitrarily large. We know that for [tex]x>0[/tex], [tex]\ln x[/tex] is continuous. This means we can pass the limit through the logarithm:
so now we're only concerned with the limit of a rational function. The leading terms in the numerator and denominator both have the same power, so we only need to consider their coefficients. In other words,