Only the letter g) and I know it’s infinity - infinity but I don’t know how to modify it to be able to find the answer

Accepted Solution

You have to use some properties of logarithms. I'm assuming the logarithm is real-valued, in which case the following hold:

[tex]\ln a-\ln b=\ln\dfrac ab[/tex]
[tex]\ln a^n=n\ln a[/tex]

for real [tex]a,b>0[/tex] and all real [tex]n[/tex].

So we can write


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,


when [tex]x\neq0[/tex], and so the limit is the same as