For a,b0, let f(x)=array|c|c| ( ((a+1)x)+b x)/(x), & x0 array be a continuous function at…

Question

For $a,b>0$, let $f(x)=\begin{array}{|c|c|} \frac{	an ((a+1)x)+b	an x}{x}, & x0 \end{array}$
be a continuous function at $x=0$. Then $\frac{b}{a}$ is equal to :

Accepted Answer

$f(x)=\begin{gathered} \frac{	an ((a+1)x)+b	an x}{x}, x0 \end{gathered}$

$f(x)$ is continuous at $x=0$

$\begin{gathered} \Rightarrow \underset{x	o {0}^{-}}{lim}f(x)=f(0)=\underset{x	o {0}^{+}}{lim}f(x) \ \underset{x	o {0}^{-}}{lim}f(x)=3 \ \Rightarrow \underset{x	o {0}^{-}}{lim}\frac{	an ((a+1)x)+b+ax}{x}=3 \ \Rightarrow a+1+b=3 \ \Rightarrow a+b=2\, \end{gathered}$

also, $\underset{x	o {0}^{+}}{lim}\frac{\sqrt{ax+{b}^{2}{x}^{2}}-\sqrt{ax}}{b\sqrt{a}x\sqrt{x}}=3$

$\begin{gathered} =\underset{x	o {0}^{+}}{lim}\frac{\sqrt{ah+{b}^{2}{h}^{2}}-\sqrt{ah}}{b\sqrt{a}	imes h\sqrt{h}}=3 \ =\underset{h	o 0}{lim}\frac{\sqrt{a+{b}^{2}{h}^{2}}-\sqrt{a}}{b\sqrt{a}h}	imes \frac{\sqrt{a+{b}^{2}h}+\sqrt{a}}{\sqrt{a+{b}^{2}h}+\sqrt{a}}=3 \ =\underset{h	o 0}{lim}\frac{a+{b}^{2}h-a}{b\sqrt{a}h(\sqrt{a+{b}^{2}h}+\sqrt{a})}=3 \ \Rightarrow \frac{{b}^{2}}{b\sqrt{a}(2\sqrt{a})}=3 \ \Rightarrow \frac{b}{2a}=3 \ \Rightarrow \frac{b}{a}=6\, \end{gathered}$

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