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\(\right\))=\(\frac\)15\(\left\)(\(\sqrt{225+\frac{5U}{\pi}\)}-15\(\right\))$. Calculate ${\(\displaystyle\]\lim\)_{U\(\to\)0^{+}}}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 $\underset{x\to 3}{\lim }\left(2x\underset{x\to 0}{\lim }f\left(x\right)\right)=-12$.