lim (2(lny)^2 + lny − 1)/(2 lny − 1)
y→√e
Write (2(lny)^2 + lny − 1)/(2 lny − 1) as (2lny - 1)(lny + 1)/(2 lny - 1), now this becomes just lny + 1.
Plugging in the limit, we have ln(√e) + 1 = [1/2 * ln(e)] + 1 = 1/2 + 1 = 3/2
really?
2(lny)^2 + lny - 1 = (2lny - 1)(lny + 1)
so your fraction is just
lny + 1
for all y ≠ √e
so, the limit is just ln(√e) + 1 = 3/2