a.
\(\lim\limits_{x\rightarrow+\infty}\left(x^4-x^2+x-1\right)=\lim\limits_{x\rightarrow+\infty}x^4\left(1-\dfrac{1}{x^2}+\dfrac{1}{x^3}-\dfrac{1}{x^4}\right)\)
Do \(\lim\limits_{x\rightarrow+\infty}x^4=+\infty\)
\(\lim\limits_{x\rightarrow+\infty}\left(1-\dfrac{1}{x^2}+\dfrac{1}{x^3}-\dfrac{1}{x^4}\right)=1>0\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}x^4\left(1-\dfrac{1}{x^2}+\dfrac{1}{x^3}-\dfrac{1}{x^4}\right)=+\infty\)
b.
\(\lim\limits_{x\rightarrow-\infty}\left(-2x^3+3x^2-5\right)=\lim\limits_{x\rightarrow-\infty}x^3\left(-2+\dfrac{3}{x}-\dfrac{5}{x^3}\right)\)
Do \(\lim\limits_{x\rightarrow-\infty}x^3=-\infty\)
\(\lim\limits_{x\rightarrow-\infty}\left(-2+\dfrac{3}{x}-\dfrac{5}{x^3}\right)=-2< 0\)
\(\Rightarrow\lim\limits_{x\rightarrow-\infty}x^3\left(-2+\dfrac{3}{x}-\dfrac{5}{x^3}\right)=+\infty\)
c.
\(\lim\limits_{x\rightarrow-\infty}\sqrt{x^2-2x+5}=\lim\limits_{x\rightarrow-\infty}\left|x\right|\sqrt{1-\dfrac{2}{x}+\dfrac{5}{x^2}}\)
Do \(\lim\limits_{x\rightarrow-\infty}\left|x\right|=+\infty\)
\(\lim\limits_{x\rightarrow-\infty}\sqrt{1-\dfrac{2}{x}+\dfrac{5}{x^2}}=1>0\)
\(\Rightarrow\lim\limits_{x\rightarrow-\infty}\left|x\right|\sqrt{1-\dfrac{2}{x}+\dfrac{5}{x^2}}=+\infty\)
d.
\(\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt{x^2+1}+x}{5-2x}=\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt{1+\dfrac{1}{x^2}}+1}{\dfrac{5}{x}-2}=\dfrac{2}{-2}=-1\)
