`underset(x to a)(Lim) ((a+2x)^((1)/(3)) - (3x)^((1)/(3)))/((3a+x)^((1)/(3)) - (4x)^((1)/(3))) (a ne 0)` is equal to :

Options

A. `((2)/(9))^(1/3)`
B. `((2)/(9))((2)/(3))^((1)/(3))`
C. `((2)/(3))^((4)/(3))`
D. `((2)/(3))((2)/(9))^((1)/(3))`

Answer

To solve the limit

\[ \lim_{x \to a} \frac{(a + 2x)^{\frac{1}{3}} - (3x)^{\frac{1}{3}}}{(3a + x)^{\frac{1}{3}} - (4x)^{\frac{1}{3}}} \]

we will follow these steps:

### Step 1: Substitute \(x = a\)

First, we substitute \(x = a\) into the limit expression to check if it results in an indeterminate form.

\[ = \frac{(a + 2a)^{\frac{1}{3}} - (3a)^{\frac{1}{3}}}{(3a + a)^{\frac{1}{3}} - (4a)^{\frac{1}{3}}} \]

This simplifies to:

\[ = \frac{(3a)^{\frac{1}{3}} - (3a)^{\frac{1}{3}}}{(4a)^{\frac{1}{3}} - (4a)^{\frac{1}{3}}} = \frac{0}{0} \]

Since we have a \(0/0\) form, we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule

According to L'Hôpital's Rule, we differentiate the numerator and the denominator separately.

**Numerator:**
Let \(f(x) = (a + 2x)^{\frac{1}{3}} - (3x)^{\frac{1}{3}}\)

Differentiating \(f(x)\):

\[ f'(x) = \frac{2}{3}(a + 2x)^{-\frac{2}{3}} \cdot 2 - \frac{1}{3}(3x)^{-\frac{2}{3}} \cdot 3 \]

This simplifies to:

\[ = \frac{4}{3}(a + 2x)^{-\frac{2}{3}} - (x)^{-\frac{2}{3}} \]

**Denominator:**
Let \(g(x) = (3a + x)^{\frac{1}{3}} - (4x)^{\frac{1}{3}}\)

Differentiating \(g(x)\):

\[ g'(x) = \frac{1}{3}(3a + x)^{-\frac{2}{3}} - \frac{4}{3}(4x)^{-\frac{2}{3}} \]

### Step 3: Rewrite the limit using derivatives

Now we can rewrite the limit as:

\[ \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

### Step 4: Substitute \(x = a\) again

Substituting \(x = a\) into the derivatives:

For the numerator:

\[ f'(a) = \frac{4}{3}(a + 2a)^{-\frac{2}{3}} - (a)^{-\frac{2}{3}} = \frac{4}{3}(3a)^{-\frac{2}{3}} - (a)^{-\frac{2}{3}} \]

For the denominator:

\[ g'(a) = \frac{1}{3}(3a + a)^{-\frac{2}{3}} - \frac{4}{3}(4a)^{-\frac{2}{3}} = \frac{1}{3}(4a)^{-\frac{2}{3}} - \frac{4}{3}(4a)^{-\frac{2}{3}} = -\frac{3}{3}(4a)^{-\frac{2}{3}} = -\frac{1}{(4a)^{\frac{2}{3}}} \]

### Step 5: Evaluate the limit

Now we have:

\[ \lim_{x \to a} \frac{\frac{4}{3}(3a)^{-\frac{2}{3}} - (a)^{-\frac{2}{3}}}{-\frac{1}{(4a)^{\frac{2}{3}}}} \]

This simplifies to:

\[ = \frac{4(3a)^{-\frac{2}{3}} - 3(a)^{-\frac{2}{3}}}{-3(4a)^{-\frac{2}{3}}} \]

### Final Answer

After simplifying, we find the limit evaluates to:

\[ \frac{1}{3} \]