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Question Number 82699 by jagoll last updated on 23/Feb/20

$$\underset{x\to 1}{\lim }\frac{\sqrt{ax-a+b}-3}{x-1}=-\frac{3}{2}$$$$findaandbwithout{L}^{\prime }hopitalrule$$

Commented by john santu last updated on 23/Feb/20

$$\left(1\right)limitmustbe\frac{0}{0}$$$$\Rightarrow \sqrt{b}=3,b=9$$$$\left(2\right)\underset{x\to 1}{\lim }\frac{\sqrt{a(x-1)+9}-3}{x-1}=-\frac{3}{2}$$$$letx-1=t\Rightarrow \underset{t\to 0}{\lim }\frac{\sqrt{at+9}-3}{t}=-\frac{3}{2}$$$$\underset{t\to 0}{\lim }\frac{at}{t}\times \underset{t\to 0}{\lim }\frac{1}{\sqrt{at+9}+3}=-\frac{3}{2}$$$$a\times \frac{1}{6}=-\frac{3}{2}\Rightarrow a=-9$$

Commented by jagoll last updated on 23/Feb/20

$$thanks$$

Commented by mathmax by abdo last updated on 23/Feb/20

$$\Rightarrow {lim}_{x\to 1}\frac{\sqrt{ax-a+b}-3}{x-1}+\frac{3}{2}=0\Rightarrow$$$${lim}_{x\to 1}\frac{2\sqrt{ax-a+b}-6+3x-3}{2(x-1)}=0\Rightarrow$$$${lim}_{x\to 1}u\left(x\right)=0and{lim}_{x\to 1}{u}^{{}^{\prime }}\left(x\right)=0with$$$$u\left(x\right)=2\sqrt{ax-a+b}+3x-9$$$${lim}_{x\to 1}u\left(x\right)=0\Rightarrow 2\sqrt{b}-6=0\Rightarrow \sqrt{b}=3\Rightarrow b=9$$$$u^{{}^{\prime }}\left(x\right)=\frac{a}{\sqrt{ax-a+b}}+3$$$${lim}_{x\to 1}{u}^{{}^{\prime }}\left(x\right)=0\Rightarrow \frac{a}{\sqrt{b}}+3=0\Rightarrow \frac{a}{3}+3=0\Rightarrow \frac{a}{3}=-3\Rightarrow a=-9$$

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