$\text { Determine whether the improper integral } \int _ { 0 } ^ { 2 } \frac { 3 } { x } d x \text { diverges or converges. Evaluate the }$ integral if it converges.
A) 3
B) 2
C) 1
D) $\frac { 2 } { 3 }$
E) diverges

Question

$\text { Determine whether the improper integral } \int _ { 0 } ^ { 2 } \frac { 3 } { x } d x \text { diverges or converges. Evaluate the }$ integral if it converges. 
A) 3
B) 2
C) 1
D) $\frac { 2 } { 3 }$
E) diverges

Quizplus · Accepted Answer

The integral in question is an improper integral since the function $\frac{3}{x}$ is not defined at $x=0$. To determine whether this integral converges or diverges, we need to compute the limit $$\lim_{t	o 0^+} \int_t^2 \frac{3}{x} dx.$$ This limit can be evaluated using the natural logarithm function: $$\lim_{t	o 0^+} \int_t^2 \frac{3}{x} dx = \lim_{t	o 0^+} \left[3 \ln|x|ight]_t^2 = 3\ln 2 - \lim_{t	o 0^+} 3\ln|t|.$$ The second term in this expression diverges to $-\infty$ as $t	o 0^+$, since the logarithm of a positive number close to zero is a large negative number. Therefore, the limit of the entire integral diverges, and the answer is E.