19. Giới hạn đã cho hữu hạn khi và chỉ khi \(a=1\)
Khi đó:
\(\lim\limits_{x\rightarrow+\infty}\left(x-\sqrt{x^2+bx-2}\right)=\lim\limits_{x\rightarrow+\infty}\dfrac{x^2-\left(x^2+bx-2\right)}{x+\sqrt{x^2+bx-2}}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{-bx+2}{x+\sqrt{x^2+bx-2}}=\lim\limits_{x\rightarrow+\infty}\dfrac{-b+\dfrac{2}{x}}{1+\sqrt{1+\dfrac{b}{x}-\dfrac{2}{x^2}}}=-\dfrac{b}{2}\)
\(\Rightarrow-\dfrac{b}{2}=3\Rightarrow b=-6\Rightarrow a+b=1+\left(-6\right)=-5\)
20.
\(\lim\limits_{x\rightarrow+\infty}\left(x^3+2x-1\right)=\lim\limits_{x\rightarrow+\infty}x^3\left(1+\dfrac{2}{x^2}-\dfrac{1}{x^3}\right)=+\infty.1=+\infty\)
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