Assess the validity of the following statement and provide a reasoning. Assume $d$ and $P$ are finite numbers.

If ${\displaystyle\lim_{z\to d}k\left(z\right)=P}$, then $k\left(d\right)=P$.

The radius of a right cylinder having a height of $15\text{ cm}$ and a surface area of $U\text{ cm}^2$ is given as $r\left(U\right)=\frac15\left(\sqrt{225+\frac{5U}{\pi}}-15\right)$. Calculate ${\displaystyle\lim_{U\to0^{+}}}r\left(U\right)$ and provide an interpretation.

Evaluate the limit.

$\displaystyle \lim_{x \to 4}{\frac{1}{x-4}\left(\frac{1}{\sqrt{x+5}}-\frac{1}{3}\right)}$

Evaluate the limit as $x\to\pm\infty$ and identify any horizontal asymptotes for the function $f\left(x\right)=\frac{x^4-25}{x^2-25}$.

Evaluate the limit of the function $f\left(x\right)={\left(\frac{5x-9}{x}\right)}^4$ as $x\to \infty$.

Use the following theorem to evaluate $\underset{x\to 0}{\lim }\frac{\sin 39x}{9x}$:

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

Find the value of $\underset{x\to 0}{\lim }f\left(x\right)$ given that $\underset{x\to 3}{\lim }\left(2x\underset{x\to 0}{\lim }f\left(x\right)\right)=-12$.