Prove Divergence/Convergence of Integral dx/ln(x) (0-1)

[Dell]

what is a similar function i can use to prove divergence/convergence of this function?
i am given the following, and asked if it converges or diverges

integral or dx/ln(x) (from 0-1)

what i did was look for a similar function, either something that is bigger and converges of smaller and diverges,
but i came across a whole bunch of problems, other than the usual, in this case my function is negative, which isn't too serious as i can take its abs value, but i have problem areas at both limits of the integral, 0 and 1, and i cannot find any function to prove its behaviour with, please help me!

someone told me to use the fact that ln(x) goes to -infinity as x goes to 0 at the same rate that e^x goes to infinity as x goes to infinity, not that i fully understand the meaning of that but i though of maybe using

g(x)=1/(e^(1/x)) so that when x=0 both g(x) and f(x) are 0 but that didnt relly help me at all

 

[lippyka]

One approach would be to consider the limit at infinity for this function. You can show that the limit of dx/ln(x) as x approaches infinity is 0, which implies convergence. You can also use the limit comparison test to compare this to another known convergent or divergent series. For example, you could compare it to 1/x, which is a known divergent series, and thus conclude that dx/ln(x) is divergent. Alternatively, you could compare it to 1/x^2, which is a known convergent series, and thus conclude that dx/ln(x) is convergent.