Determine a region whose area is equal to the given limit. Do not evaluate the limit. lim n → ∞ ni = 1𝜋3n tani𝜋3n x tan(x) on − 𝜋3, 𝜋3tan(x) on [0, 3𝜋]    x tan(x) on 0, 𝜋3tan(x) on − 𝜋3, 𝜋3tan(x) on 0, 𝜋3

Determine a region whose area is equal to the given limit. Do not evaluate the limit.lim n→∞ n4n1 + 4ini = 1

(a) Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit.lim n→∞ ni = 1

Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.f(x) = x2 + 1 + 2x,    2 ≤ x ≤ 4lim n → ∞ ni = 1

limx→0+(tan (2x))x

Find the vertical asymptotes (if any) of the graph of the function. (Use n as an arbitrary integer if necessary. If an answer does not exist, enter DNE.)f(x) =  tan(πx)