a: \(\lim\limits_{x\rightarrow1}\dfrac{\sqrt{x^2+x+7}-3}{x-1}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{x^2+x+7-9}{\left(x-1\right)\left(\sqrt{x^2+x+7}+3\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x+2\right)\left(x-1\right)}{\left(x-1\right)\left(\sqrt{x^2+x+7}+3\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{x+2}{\sqrt{x^2+x+7}+3}=\dfrac{1+2}{\sqrt{1+1+7}+3}=\dfrac{3}{3+3}=\dfrac{3}{6}=\dfrac{1}{2}\)
b: \(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+5x}-x\right)\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{x^2+5x-x^2}{\sqrt{x^2+5x}+x}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{5x}{\sqrt{x^2+5x}+x}=\lim\limits_{x\rightarrow+\infty}\dfrac{5}{\sqrt{1+\dfrac{5}{x}}+1}=\dfrac{5}{2}\)
c: \(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+5x}+x\right)\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{x^2+5x-x^2}{\sqrt{x^2+5x}-x}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{5x}{\sqrt{x^2+5x}-x}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{5}{\sqrt{1+\dfrac{5}{x}}-1}=+\infty\)
d: \(\lim\limits_{x\rightarrow3}\dfrac{x^2-x-6}{\left|x-3\right|}\)
\(=\lim\limits_{x\rightarrow3}\dfrac{\left(x-3\right)\left(x+2\right)}{\left|x-3\right|}\)
\(=\lim\limits_{x\rightarrow3}\dfrac{\left(x+2\right)}{\pm1}\)
\(=\left[{}\begin{matrix}3+2=5\\-3-2=-5\end{matrix}\right.\)
