Limits

Polynomials are continuous, so the limit equals the value: $3(2)+1 = 7$.

One of the two standard trigonometric limits, valid with $x$ in radians. For small angles $\sin x \approx x$.

Substituting gives $\frac00$. Factorising gives $\dfrac{(x-2)(x+2)}{x-2} = x+2$, which tends to 4.

The degrees match, so the limit is the ratio of the leading coefficients: $\dfrac42 = 2$.

Rewrite as $\dfrac43 \cdot \dfrac{\sin 4x}{4x}$. The second factor tends to 1, leaving $\dfrac43$.

$\dfrac{(x+h)^3 - x^3}{h} = \dfrac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2$, which tends to $3x^2$ as $h \to 0$.