Understanding the Problem

We are given a quadratic equation in x, with parameter a > -1:

$(\sqrt[3]{1+a}-1)x^2+(\sqrt{1+a}-1)x+(\sqrt[6]{1+a}-1)=0$

Let α(a) and β(a) be its roots. We need to find the limits of these roots as a approaches 0 from the positive side (a → 0⁺).

Step 1: Analyze the Coefficients as a → 0⁺

As a → 0, all the terms (1+a)^(1/n) - 1 approach 0. This makes the equation approach 0 = 0, which is indeterminate. To find the roots in this limit, we must simplify the equation by dividing by a common factor that also tends to 0.

Let's define the coefficients:

• A = $\sqrt[3]{1+a}-1$
• B = $\sqrt{1+a}-1$
• C = $\sqrt[6]{1+a}-1$

Notice that all coefficients are of the form (1+a)^(1/n) - 1. We can use the binomial approximation for small a: (1+a)^(1/n) ≈ 1 + a/n. Therefore:

• A ≈ a/3
• B ≈ a/2
• C ≈ a/6

So, our equation becomes approximately:

$\frac{a}{3}{x}^2+\frac{a}{2}x+\frac{a}{6}=0$

Since a → 0⁺, we can divide the entire equation by a (which is not zero) to simplify:

$\frac{1}{3}{x}^2+\frac{1}{2}x+\frac{1}{6}=0$

Step 2: Solve the Simplified Quadratic Equation

To eliminate fractions, multiply the entire equation by 6 (the LCM of 3, 2, and 6):

$6\times (\frac{1}{3}{x}^2+\frac{1}{2}x+\frac{1}{6})=6\times 0$

$2x^2+3x+1=0$

Now, we solve this quadratic equation for x.

Using the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where a=2, b=3, c=1.

Discriminant, D = b² - 4ac = (3)² - 4(2)(1) = 9 - 8 = 1

Therefore, the roots are:

$x=\frac{-3\pm \sqrt{1}}{2\times 2}=\frac{-3\pm 1}{4}$

So, the two roots are:

$x_1=\frac{-3+1}{4}=\frac{-2}{4}=-\frac{1}{2}$

$x_2=\frac{-3-1}{4}=\frac{-4}{4}=-1$

Step 3: Identify the Limits

As a → 0⁺, the roots α(a) and β(a) of the original equation approach the roots of the simplified equation we just solved.

Therefore:

$\underset{a\to 0^{+}}{\lim }\alpha \left(a\right)=-\frac{1}{2}$

$\underset{a\to 0^{+}}{\lim }\beta \left(a\right)=-1$

(The order of the roots, i.e., which one is α and which one is β, is arbitrary and depends on convention. The pair of limits is what matters.)

Final Answer

The limits are $\frac{-1}{2}$ and $-1$.

Related Topics & Formulae

Key Concepts

• Limits: The value that a function (or a sequence) "approaches" as the input (or index) "approaches" some value.
• Indeterminate Forms: Expressions like 0/0 that require further analysis to find a limit.
• Binomial Approximation: For small x, (1+x)^n ≈ 1 + nx. This is the first-order Taylor expansion.
• Roots of a Quadratic Equation: The solutions to ax² + bx + c = 0 are given by the quadratic formula.

Important Formulae

• Quadratic Formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
• Binomial Approximation: ${(1+y)}^k\approx 1+ky,\text{ for }\left|y\right|\ll 1$