📘 calculus and algebra

1. Problem statement: Solve the cooling, hoop fluctuation, optimization and differentiation problems as given. 1. (a) Coffee model: The temperature of the coffee is modeled by $T(t

1. **Evaluate the limits at infinity and zero:** - $L_1 = \lim_{x \to \pm \infty} \frac{x^2 - 8}{3x^2 - 2}$

1. **Evaluate the indefinite integral** \( \int \frac{1}{(x+1)^2} \, dx \). Use substitution: let \( u = x + 1 \), so \( du = dx \).

1. **Sketch and analyze** $f(x)=\frac{x^2 - 81}{x + 9}$. Simplify numerator: $x^2 - 81 = (x - 9)(x + 9)$. The function simplifies to $f(x) = x - 9$ for $x \neq -9$ (since denominat

1. **مسألة القيم الحدية:** (a)(i) إيجاد: $$\lim_{x \to 0} \left(1 - \left(\frac{3x}{5}\right)^{\frac{5}{3}}\right)$$