Normalizing a wave packet - cannot understand the solution

[physiker99]

This question is about normalizing a wave packet, this is actually the solution and I couldn't understand 2 points.

- I cannot see how the red encircled part do not diverge to infinity.

- And I cannot understand how the very last line is derived from the 2nd last one.

For the second one I tried to call i(po-px) as A and 1/(deltaX) B but that did not lead anywhere.

 

[zhermes]

For the red box
$$e^{i \alpha x - \beta x} = e^{i \alpha x}e^{-\beta x}$$
for x -> infinity: the first term oscillates between 1 and -1 (in accordance with euler's equation), but the second term approaches zero. So the whole thing approaches zero, and you're just left with the evaluation at x = 0.And for the penultimate to ultimate lines, you just have to get a common denominator to combine the fractions; a bunch of stuff will cancel... you multiply through by delta X / delta X