Consider the differential equation \( \frac{dy}{dx} = 4x^2 \) with the initial condition \( y(1) = 5 \). Determine the specific solution.

1. \( \lim _{x \rightarrow 0} \frac{\arcsin 5 x}{x^{2}-x} \)

Calcolare il limite \( \lim _{x \rightarrow+\infty} x \log \left(1-\frac{1}{x}\right) \) amente all'esercizio 8.18 , si trova il valore limite -1\( ] \)

The temperature of a cube of aluminium (decreases from \( 120^{\circ} \mathrm{C} \) to \( T^{\circ} \mathrm{C} \) ) following the equation \( T=100(0.95)^{x} \) after \( x \) seconds. calculate (i) suhu aluminium selepas 4 saat. the temperature of the aluminium cube after 4 seconds. \( T=100(0.95)^{4} \) \( T=81.45^{\circ} \) (ii) masa, \( t \) dalam saat untuk suhu aluminium menyusut daripada \( 120^{\circ} \mathrm{C} \) Do kepada \( 90^{\circ} \mathrm{C} \). conection the time, \( t \) in seconds for the aluminium to decreases from \( 120^{\circ} \mathrm{C} \) to \( 90^{\circ} \mathrm{C} \).

\( \lim _ { x \rightarrow - \infty } ( 2 - x ) \sin \frac { 1 } { x } \)

\( \sum _ { 0 } ^ { \infty } ( - 1 ) ^ { n } x ^ { 4 n + 2 } \)