These are all L'Hopital's limits, so we must use a method of taking natural logartihms, taking derivatives of the top and bottom of a fraction, etc.
lim as x approaches 0 of (e^x + x)^(1/x)
ln (e^x + x)^(1/x)
(1/x)(ln (e^x + x))
(1/x)ln (e^x) + (1/x)(ln x)
1 + (1/x)(ln x)
1 + (1/x)
before i continue, i must apologize because I dont know how to make my posts be in a more nice format where the notation is all correct and what not. my apologies.
lim x->∞ ((ln x)^(1/x))
y= (ln x)^(1/x)
ln y = (ln(lnx))/x
= 1/(x(ln x))
= x/ (x + ln x)
and that definitely cant be right
I also should explain a bit. I was taught to set the function that the limit is being taken of equal to y, taking the ln of both sides, bringing the exponent down as a coefficient, and then in these two cases you arrive upon a fraction, so you derive the numerator and denominator of the fraction until the limit is able to be evaluated, the i can setup the limit again as the ln of the original limit being equal to the value obtained in the previous step, and then raising each side to the power of e and getting the correct answer. This is what I did for the other problems in the set and I got all the right answers. I'm just getting stuck on these ones :/