Equation: 5e^(2x) + 3e^x - 2 = 0
Let y = e^x , 5y² + 3y - 2 = 0
Solve quadratic: y = [-3 ± sqrt(9 + 40)]/10 = [-3 ± 7]/10
y = (4)/10 = 0.4, y = (-10)/10 = -1 , discard
e^x = 0.4 , x = ln(0.4) = -0.916

lim_{x ,0} (x³ sin5) / 5³ = sin5 / 125
lim_{x ,0} (x³ sin5x) / 5x = lim_{x ,0} (x² sin5x) / 5 = lim x ,0 x² × (5x/5) ? Use standard: sin(5x)/5x 1
lim = x² , 0

Equation: 4x³ + 6x² + 4x + 1 = 0
Try factoring: (2x + 1)³ = 8x³ + 12x² + 6x +1 , divide by 2? Check
Use cubic formula or factor by trial: 4x³ + 6x² + 4x +1 = (2x+1)(2x² + 2x +1) = 0
2x +1 =0 , x = -1/2
2x² +2x +1=0 , x = [-2 ± sqrt(4 -8)]/4 = (-2 ± sqrt(-4))/4 = (-2 ± 2i)/4 = -1/2 ± i/2
x = -1/2, -1/2 + i/2, -1/2 - i/2

Series: (n / n!) x
Ratio test: L = lim |a / a |
a = n / n!
a / a = ((n+1) ¹ / (n+1)!) × (n! / n ) = ((n+1) ¹ / n ) × (1/(n+1)) = ((n+1) / n ) = (1 + 1/n)n e
Radius R = 1 / L = 1 / e

y = x^x ln y = x ln x
d/dx(ln y) = d/dx(x ln x) (1/y) dy/dx = ln x + 1
dy/dx = y (ln x + 1) = x^x (ln x + 1)