Finding Limits at Infinity

Find:

1. \(\lim\limits_{x\rightarrow -\infty }\dfrac{4}{x^{2}}\)
2. \(\lim\limits_{x\rightarrow \infty }\left( -\dfrac{10}{\sqrt{x}} \right)\)
3. \(\lim\limits_{x\rightarrow \infty }\left( 2+\dfrac{3}{x}\right)\)

Solution (a)\(\lim\limits_{x\rightarrow -\infty }\dfrac{4}{x^{2}}=4\left(\lim\limits_{x\rightarrow -\infty }\dfrac{1}{x}\right)^2=4\,{\cdot}\,0=0\)

(b)\(\lim\limits_{x\rightarrow \infty }\left( -\dfrac{10}{\sqrt{x}}\right) =-10\lim\limits_{x\rightarrow \infty }\dfrac{1}{\sqrt{x}}=-10\,{\cdot}\,\lim\limits_{x\rightarrow \infty }\sqrt{\dfrac{1}{x}}=-10\,{\cdot}\,\sqrt{\lim\limits_{x\rightarrow \infty }\dfrac{1}{x}}\) \(=-10\,{\cdot}\,0=0\)

(c) \(\lim\limits_{x\rightarrow \infty }\left( 2+\dfrac{3}{x}\right) =\lim\limits_{x\rightarrow \infty }2+\lim\limits_{x\rightarrow \infty } \dfrac{3}{x}=2+3\cdot \lim\limits_{x\rightarrow \infty }\dfrac{1}{x} =2+3\cdot 0=2\)