Find the following limit

Question

Find the following limit

lim tan^3(2x)sin^2(5x)

x-->0 -------------------------- (division symbol)

(2x)^5

Please provide work along with answer and answer ASAP. Thank you so much.

Answer 1

limx tan³(2x)sin²(5x)/(2x)⁵ = lim (sin2x/2x)³· (sin5x/5x)²·5²/(2²cos³2x) = 1³·1²·25/(4·1³) = 25/4

x→0 x→0

Answer 2

Using first term of series expansion for tangent and sine

tan³(2x) ≈ (2x)³ = 8x³

sin²(5x) ≈(5x)² = 25x²

lim (8•25x⁵/(32x⁵) = 25/4 = 6.25

x→0