Evaluate `lim_(xto0) ((x+2)^(1//3)-2^(1//3))/(x)`

Options

A. `1/(3)(2)^(2//3)`
B. `1/(3.(2)^(2//3))`
C. `1/(3)`
D. none of the above

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{(x+2)^{1/3} - 2^{1/3}}{x}, \]

we first check the form of the limit by substituting \(x = 0\):

\[ \frac{(0 + 2)^{1/3} - 2^{1/3}}{0} = \frac{2^{1/3} - 2^{1/3}}{0} = \frac{0}{0}. \]

This is an indeterminate form, so we need to manipulate the expression to resolve it.

### Step 1: Rewrite the limit
We can rewrite the limit as:

\[ \lim_{x \to 0} \frac{(x + 2)^{1/3} - 2^{1/3}}{x} = \lim_{x \to 0} \frac{(x + 2)^{1/3} - 2^{1/3}}{(x + 2) - 2} \cdot \frac{(x + 2) - 2}{x}. \]

### Step 2: Apply the standard limit result
Now we can apply the standard limit result:

\[ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a), \]

where \(f(x) = (x + 2)^{1/3}\) and \(a = 2\). We need to find \(f'(x)\).

### Step 3: Differentiate \(f(x)\)
Using the power rule, we differentiate \(f(x)\):

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

### Step 4: Evaluate the derivative at \(x = 2\)
Now we evaluate \(f'(2)\):

\[ f'(2) = \frac{1}{3}(2 + 2)^{-2/3} = \frac{1}{3}(4)^{-2/3} = \frac{1}{3} \cdot \frac{1}{4^{2/3}}. \]

Since \(4^{2/3} = (2^2)^{2/3} = 2^{4/3}\), we have:

\[ f'(2) = \frac{1}{3 \cdot 2^{4/3}}. \]

### Step 5: Simplify the expression
Thus, we can write:

\[ \lim_{x \to 0} \frac{(x + 2)^{1/3} - 2^{1/3}}{x} = f'(2) = \frac{1}{3 \cdot 2^{4/3}} = \frac{1}{3} \cdot 2^{-4/3} = \frac{1}{3 \cdot 2^{4/3}}. \]

### Final Answer
So, the limit evaluates to:

\[ \frac{1}{3 \cdot 2^{4/3}}. \]