Calculus Answers

Find limit using L' Hopitals rule:

(a) lim x "\\to" 0 (cos2x)^(1/x^2)

Evaluate the limit as x turns to -∞

Lim [3x + √(9x²-Ix-2I)]

The domain of the function define by f(x) = √[Inx/ (Inx-1)] is?

Simplify the sum Σ from r=0 to 2n+1 of [3/(r+1)(r+2)]

Calculate the volume created by revolution around the axes OX, of D of the plane (OXY) where D={(x,y) ∈R: 0 ≤ y ≤x, (x-2)² +y² ≤ 4}

Evaluate the limit as x turns to 2⁺

lim {sin[In (x-1)] / √(x-2)}

Evaluate ∫₀^π √(1+cos8xdx)

The sum of the third and tenth term of an AP is 12. Given that the sum of the 6th terms is 3 less than one third of the eight term. Find the first term and the common difference.

Determine the monotony of the following sequences.

a) Un = 2n+3 / 5n-1 n € N

b) Un = √(n+3) - √(n+1) n € N^*

c) Un = Σ from r=1 to n (1/ n+r) , n € N

Find the area of the region bounded by the curves y= -x²/3 + 2/3x + 5 and if -x+5.

Find also the volume when the area is rotated completely about the x-axis.