a) $\lim\dfrac{2 - 5^{n+2}}{2^{2n} + 1+ 2.5^{n-1}}$
$=\lim\dfrac{10 - 125.5^n}{5.4^n + 5 + 2.5^n}$
$=\lim\dfrac{10\left(\dfrac15\right)^n - 125}{5.\left(\dfrac45\right)^n + 5.\left(\dfrac15\right)^n + 2}$
$=\dfrac{0 - 125}{0 + 0 + 2}$
$= -\dfrac{125}{2}$
b) $\lim\dfrac{2^n + 3^{2n}}{2^{n+3} + 4.7^n}$
$=\lim\dfrac{2^n + 9^n}{8.2^n + 4.7^n}$
$=\lim\dfrac{\left(\dfrac27\right)^n + \left(\dfrac97\right)^n}{8.\left(\dfrac27\right)^n + 4}$
$=\dfrac{0 + \infty}{0 + 4}$
$= +\infty$
