Calculate the limit of an integral

Question

How can I use Mathematica to find the limit:$$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}=?$$

I tried this Limit[Integrate [(E^(-y*x) - 1)/y^3, {y, 1, \[Infinity]}]/Log[1 + x], x -> 0, Direction -> 1], but the returned answer isUndefined. Any suggestions ?

Answer 1

$Version

"10.2.0 for Mac OS X x86 (64-bit) (July 7, 2015)"

The Direction of the limit should be -1 to approach 0+. From the documentation, "Direction -> -1 takes variables to approach their limits by decreasing from larger values."

Limit[Integrate[(E^(-y*x) - 1)/y^3, {y, 1, \[Infinity]}], x -> 0, 
 Direction -> -1]

0

Including the 1/Log[1+x] factor

Limit[Integrate[(E^(-y*x) - 1)/y^3, {y, 1, \[Infinity]}]/Log[1 + x], 
 x -> 0, Direction -> -1]

-1