what could this limit be?

Question

lim (1+8x)^(cscx)
x->0

Answer 1

c = lim[(1+8x)^(cscx)]
Take ln of both sides
ln c = ln [lim (1+8x)^(cscx)]
= lim [ csc(x) ln (1+ 8x)]
= lim[ ln(1 +8x)/ sin(x)]
Using l'hopital's rule because plugging in 0 for x gives 1/0.
Taking derivatives of numerator and denominator gives
= lim[ (8 /(1 + 8x)) / cos(x)
ln c = lim ( 8/ 1) = 8
c = exp(8) = 2980.958

Answer 2

lim (1+8*x)^(csc(x)) = e^8 ≈ 2980.96
x->0