Answer
-54
Work Step by Step
We have to determine $L=\lim\limits_{x \to 4} \dfrac{3(x-4)\sqrt{x+5}}{3-\sqrt{x+5}}$.
Multiply both numerator and denominator by the conjugate of the denominator:
$L=\lim\limits_{x \to 4} \dfrac{3(x-4)\sqrt{x+5}}{3-\sqrt{x+5}}\cdot\dfrac{3+\sqrt{x+5}}{3+\sqrt{x+5}}$
$=\lim\limits_{x \to 4} \dfrac{3(x-4)\sqrt{x+5}(3+\sqrt{x+5})}{3^2-(\sqrt{x+5})^2}$
$=\lim\limits_{x \to 4} \dfrac{3(x-4)\sqrt{x+5}(3+\sqrt{x+5})}{9-(x+5)}$
$=\lim\limits_{x \to 4} \dfrac{3(x-4)\sqrt{x+5}(3+\sqrt{x+5})}{4-x}$
Simplify:
$L=-\lim\limits_{x \to 4} 3\sqrt{x+5}(3+\sqrt{x+5})$
Determine the limit:
$L=-3\sqrt{4+5}(3+\sqrt{4+5})=-3(3)(3+3)=-54$
