\(\lim\limits_{x\rightarrow1^+}f\left(x\right)=\lim\limits_{x\rightarrow1^+}\dfrac{\sqrt{x^2-1}+\sqrt[3]{\left(x-1\right)^3}}{\sqrt{x-1}}=\lim\limits_{x\rightarrow1^+}\dfrac{\left(x^2-1\right)^{\dfrac{1}{2}}+x-1}{\left(x-1\right)^{\dfrac{1}{2}}}=\lim\limits_{x\rightarrow1^+}\dfrac{\dfrac{1}{2}\left(x^2-1\right)^{-\dfrac{1}{2}}.2+1}{\dfrac{1}{2}\left(x-1\right)^{-\dfrac{1}{2}}}\)

\(=\dfrac{1}{0}=+\infty\)

\(\lim\limits_{x\rightarrow1^-}f\left(x\right)=\lim\limits_{x\rightarrow1^-}\dfrac{\sqrt[3]{x}-1}{\sqrt{2}-\sqrt{x+1}}=\lim\limits_{x\rightarrow1^-}\dfrac{\left(x-1\right)\left(\sqrt{2}+\sqrt{x+1}\right)}{[\left(\sqrt[3]{x}\right)^2+\sqrt[3]{x}+1]\left(1-x\right)}=\lim\limits_{x\rightarrow1^-}\dfrac{-\left(\sqrt{2}+\sqrt{1+1}\right)}{1+1+1}=-\dfrac{2\sqrt{2}}{3}\)

\(f\left(1\right)=\sqrt{2}\)

\(\lim\limits_{x\rightarrow1^-}f\left(x\right)\ne\lim\limits_{x\rightarrow1^+}f\left(x\right)\ne f\left(x\right)\)=> ham gian doan tai x=1

\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{ax+1}-\sqrt[]{1-bx}}{x}=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{ax}{\sqrt[3]{\left(ax+1\right)^2}+\sqrt[3]{ax+1}+1}+\dfrac{bx}{1+\sqrt[]{1-bx}}}{x}\)

\(=\lim\limits_{x\rightarrow0}\left(\dfrac{a}{\sqrt[3]{\left(ax+1\right)^2}+\sqrt[3]{ax+1}+1}+\dfrac{b}{1+\sqrt[]{1-bx}}\right)=\dfrac{a}{3}+\dfrac{b}{2}\)

Hàm liên tục tại \(x=0\) khi:

\(\dfrac{a}{3}+\dfrac{b}{2}=3a-5b-1\Leftrightarrow8a-11b=3\)

Lời giải:

Do $-3<-1$ nên:

$f(-3)=3(-3)^2-(-3)+1=31$

Do $0> -1$ nên:

$f(0)=\sqrt{0+1}-2=-1$

$\Rightarrow f(-3)+f(0)=31+(-1)=30$

\(\lim\limits_{x\rightarrow1^+}f\left(x\right)=\lim\limits_{x\rightarrow1^+}\dfrac{\sqrt{x+3}-2}{x-1}=\lim\limits_{x\rightarrow1^+}\dfrac{x-1}{\left(x-1\right)\left(\sqrt{x+3}+2\right)}=\lim\limits_{x\rightarrow1^+}\dfrac{1}{\sqrt{x+3}+2}=\dfrac{1}{4}\)

\(f\left(1\right)=\lim\limits_{x\rightarrow1^-}f\left(x\right)=\lim\limits_{x\rightarrow1^-}\left(mx\right)=m\)

Hàm liên tục tại x=1 khi: \(\lim\limits_{x\rightarrow1^+}f\left(x\right)=\lim\limits_{x\rightarrow1^-}f\left(x\right)=f\left(1\right)\)

\(\Leftrightarrow m=\dfrac{1}{4}\)

\(\lim\limits_{x\rightarrow1^+}f\left(x\right)=\lim\limits_{x\rightarrow1^+}\dfrac{\sqrt{x+3}-2}{x-1}=\lim\limits_{x\rightarrow1^+}\dfrac{x-1}{\left(x-1\right)\left(\sqrt{x+3}+2\right)}=\lim\limits_{x\rightarrow1^+}\dfrac{1}{\sqrt{x+3}+2}=\dfrac{1}{4}\)

\(f\left(1\right)=\lim\limits_{x\rightarrow1^-}f\left(x\right)=\lim\limits_{x\rightarrow1^-}\left(ax+2\right)=a+2\)

Hàm liên tục tại x=1 khi:

\(a+2=\dfrac{1}{4}\Rightarrow a=-\dfrac{7}{4}\)

\(f\left(0\right)=2.0+m+1=m+1\)

\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[3]{x+1}-1}{x}=\lim\limits_{x\rightarrow0^+}\dfrac{x+1-1}{x(\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{x+1}+1)}=\dfrac{1}{1+1+1}=\dfrac{1}{3}\)\(f\left(0\right)=\lim\limits_{x\rightarrow0^+}f\left(x\right)\Leftrightarrow m+1=\dfrac{1}{3}\Rightarrow m=-\dfrac{2}{3}\)

\(4\in(2;5]\Rightarrow f\left(4\right)=4^2-1=15\)