The Solution:
Given the functions below:
[tex]-\ln (x)\text{ and }\frac{1}{x}[/tex]We are required to tell which of them will go to infinity faster as x tends to positive zero.
we shall examine each of the two functions by investigating their slopes under the given conditions.
[tex]\begin{gathered} T\text{he derivative of -}\ln (x)\text{ is }\frac{-1}{x}\text{ , while that of} \\ \frac{1}{x}\text{ is }\frac{-1}{x^2} \end{gathered}[/tex]Comparing the absolute values of their slopes at positive smaller values of x, (at x<1 , we get
[tex]\begin{gathered} |\frac{-1}{x^2}|>|\frac{-1}{x}| \\ \text{Clearly, the function }\frac{1}{x}\text{ goes to }\infty\text{ faster as x}\rightarrow0^+ \end{gathered}[/tex]Therefore, the correct answer is 1/x
