Is the solution I posted wrong?

For the last part, using L'Hopital's rule on (1 - xln(2))/2^x gives me -ln(2)/[ln(2) * 2^x] = -1/2^x = 0 (ignoring the limit operators - is operator the right word?).

Is the solution wrong or am I not seeing something?

Any input would be appreciated!

Thanks in advance!