a: \(\lim\limits_{x\rightarrow2}\dfrac{3x^2+2x-16}{x-2}\)
\(=\lim\limits_{x\rightarrow2}\dfrac{3x^2-6x+8x-16}{x-2}\)
\(=\lim\limits_{x\rightarrow2}\dfrac{\left(x-2\right)\left(3x+8\right)}{x-2}\)
\(=\lim\limits_{x\rightarrow2}3x+8=3\cdot2+8=6+8=14\)
b: \(\lim\limits_{x\rightarrow7}\dfrac{x^2-49}{2x-14}\)
\(=\lim\limits_{x\rightarrow7}\dfrac{\left(x-7\right)\left(x+7\right)}{2\left(x-7\right)}=\lim\limits_{x\rightarrow7}\dfrac{x+7}{2}\)
\(=\dfrac{7+7}{2}=\dfrac{14}{2}=7\)
c: \(\lim\limits_{x\rightarrow1}\dfrac{\sqrt{9x+7}-4}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{9x+7-16}{x-1}:\left(\sqrt{9x+7}+4\right)\)
\(=\lim\limits_{x\rightarrow1}\dfrac{9}{\sqrt{9x+7}+4}=\dfrac{9}{\sqrt{9+7}+4}\)
\(=\dfrac{9}{4+4}=\dfrac{9}{8}\)
d: \(\lim\limits_{x\rightarrow3}\dfrac{4x-7}{x^2-5}=\dfrac{4\cdot3-7}{3^2-5}=\dfrac{12-7}{9-5}=\dfrac{5}{4}\)
e: \(\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt{9x^2+5x}}{4x-3}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt{9+\dfrac{5}{x}}}{4-\dfrac{3}{x}}=\dfrac{\sqrt{9+0}}{4-0}=\dfrac{3}{4}\)
