For each t ∈ (0, 1), the surface Pt in ℝ3 is defined by
Pt = {(x, y, z) : (x2 + y2)z = 1, t2 ≤ x2 + y2 ≤ 1}.
Let at ∈ ℝ be the surface area of Pt. Then
1
\(a_t=\displaystyle \iint_{t^2 \leq x^2+y^2 \leq 1} \sqrt{1+\frac{4 x^2}{\left(x^2+y^2\right)^4}+\frac{4 y^2}{\left(x^2+y^2\right)^4}} \) dx dy
2
\(\displaystyle a_t=\iint_{t^2 \leq x^2+y^2 \leq 1} \sqrt{1+\frac{4 x^2}{\left(x^2+y^2\right)^2}+\frac{4 y^2}{\left(x^2+y^2\right)^2}} \) dx dy
3
the limit \(\displaystyle\lim_{t \rightarrow 0^+}\) at does NOT exist
4
the limit \(\displaystyle\lim_{t \rightarrow 0^+}\) at exists