Let alpha(a) and beta(a) be the roots of the | vedclass

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(B) The given equation is $((1+a)^{1/3}-1)x^2 + ((1+a)^{1/2}-1)x + ((1+a)^{1/6}-1) = 0$.Let $1+a = t^6$. As $a \rightarrow 0^{+}$,$t \rightarrow 1^{+}

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$-\frac{1}{2}$ and $-1$

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(B) The given equation is $((1+a)^{1/3}-1)x^2 + ((1+a)^{1/2}-1)x + ((1+a)^{1/6}-1) = 0$.Let $1+a = t^6$. As $a ightarrow 0^{+}$,$t ightarrow 1^{+}$.The equation becomes $(t^2-1)x^2 + (t^3-1)x + (t-1) = 0$.Dividing by $(t-1)$ (since $t 
eq 1$ as $a > 0$):$(t+1)x^2 + (t^2+t+1)x + 1 = 0$.Taking the limit as $t ightarrow 1$:$(1+1)x^2 + (1^2+1+1)x + 1 = 0$.$2x^2 + 3x + 1 = 0$.Factoring the quadratic equation:$(2x+1)(x+1) = 0$.Thus,the roots are $x = -\frac{1}{2}$ and $x = -1$.

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a}-1) x^2+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$ where $a > -1$. Then $\lim _{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _{a \rightarrow 0^{+}} \beta(a)$ are

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