If `lim_(x to 2) (x - 2)/(""^(3)sqrt(x) - ""^(3)sqrt(2)) = lim_(x to k) (x^(2) - k^(2))/(x - k)` find the value of K

Answer

To solve the problem, we need to evaluate the limits on both sides of the equation and find the value of \( k \).

### Step-by-Step Solution:

1. **Evaluate the Left-Hand Side Limit:**
We start with the limit:
\[ \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x} - \sqrt[3]{2}} \]
We can apply the formula for limits:
\[ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} \]
In our case, \( n = 1 \) and \( a = 2 \), so we rewrite the limit:
\[ \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x} - \sqrt[3]{2}} = \frac{1}{\frac{1}{3} \cdot 2^{\frac{1}{3}}} \]
This simplifies to:
\[ = \frac{3}{2^{\frac{1}{3}}} \]

2. **Evaluate the Right-Hand Side Limit:**
Now we evaluate the limit:
\[ \lim_{x \to k} \frac{x^2 - k^2}{x - k} \]
We can factor the numerator:
\[ x^2 - k^2 = (x - k)(x + k) \]
Thus, we have:
\[ \lim_{x \to k} \frac{(x - k)(x + k)}{x - k} = \lim_{x \to k} (x + k) = 2k \]

3. **Set the Limits Equal to Each Other:**
Now we set the two limits equal to each other:
\[ \frac{3}{2^{\frac{1}{3}}} = 2k \]

4. **Solve for \( k \):**
To find \( k \), we rearrange the equation:
\[ k = \frac{3}{2 \cdot 2^{\frac{1}{3}}} \]
This simplifies to:
\[ k = \frac{3}{2^{1 + \frac{1}{3}}} = \frac{3}{2^{\frac{4}{3}}} \]

5. **Final Result:**
Thus, the value of \( k \) is:
\[ k = \frac{3}{2^{\frac{4}{3}}} \]