What is the value of lim_{n-inf} int_0^1 (x - 1)^n * ln(x) dx ?

A. -1
B. 0
C. 1
D. infty

Question

What is the value of lim_{n->inf} int_0^1 (x - 1)^n * ln(x) dx ?

A. -1
B. 0
C. 1
D. infty

Accepted Answer

0 For x in the open interval (0, 1), the factor (x-1)^n = (-(1-x))^n has absolute value (1-x)^n which tends to 0 as n tends to infinity since 0 < 1-x < 1. The function ln(x) is integrable on (0,1). By the Dominated Convergence Theorem, we can pass the limit inside the integral: the integrand pointwise goes to 0, so the whole integral goes to 0.

What is the value of lim_{n->inf} int_0^1 (x - 1)^n * ln(x) dx ? A. -1 B. 0 C. 1 D. infty