Need help finding the value of these exponential limits and learn how to deal with them

[dusky plume]

cannot use hospital rule as it is not in our curriculum

[scenic flare]

for the 1st one you can expand it using the binomial theorem and use the formula:
lim(x->inf, 1/x^n) = 0
for the 2nd one you can factor out e from both the numerator and denominator, cancel out both of them, then you'll be left with:
(e^(x²-1)-1)/(e^(x-1)-1)
= (e^((x+1)(x-1))-1)/(e^(x-1)-1)
= ((e^(x-1))^(x+1)-1)/(e^(x-1)-1)
then use the formula of a^n-1 to expand the numerator and see if you can cancel out the denominator, then substitute x = 1

[dusky plume]

I found this