`lim_(x to 27) (x^(1//3) + 3) (X^(1//3) - 3))/(x - 27)` is equal to

Options

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

Answer

To solve the limit \( \lim_{x \to 27} \frac{(x^{1/3} + 3)(x^{1/3} - 3)}{x - 27} \), we can follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = 27 \) into the expression:
\[ x^{1/3} = 27^{1/3} = 3 \]
Thus, the numerator becomes:
\[ (3 + 3)(3 - 3) = 6 \cdot 0 = 0 \]
And the denominator becomes:
\[ 27 - 27 = 0 \]
This gives us the indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter the \( \frac{0}{0} \) form, it indicates that further manipulation is needed to resolve the limit.

### Step 2: Factor the denominator
We can rewrite the denominator \( x - 27 \) as a difference of cubes:
\[ x - 27 = x^{1/3}^3 - 3^3 \]
Using the identity \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), we can factor it as:
\[ x - 27 = (x^{1/3} - 3)(x^{2/3} + 3x^{1/3} + 9) \]

**Hint:** Recognizing the difference of cubes can help simplify the expression.

### Step 3: Rewrite the limit
Now we can rewrite the limit:
\[ \lim_{x \to 27} \frac{(x^{1/3} + 3)(x^{1/3} - 3)}{(x^{1/3} - 3)(x^{2/3} + 3x^{1/3} + 9)} \]
We can cancel the common factor \( (x^{1/3} - 3) \):
\[ \lim_{x \to 27} \frac{x^{1/3} + 3}{x^{2/3} + 3x^{1/3} + 9} \]

**Hint:** Canceling common factors simplifies the limit, making it easier to evaluate.

### Step 4: Substitute again
Now we substitute \( x = 27 \) into the simplified expression:
\[ x^{1/3} = 3 \quad \text{and} \quad x^{2/3} = 3^2 = 9 \]
So we have:
\[ \frac{3 + 3}{9 + 3 \cdot 3 + 9} = \frac{6}{9 + 9 + 9} = \frac{6}{27} \]

**Hint:** After simplification, substituting the limit value again will yield a determinate form.

### Step 5: Simplify the result
Now we simplify \( \frac{6}{27} \):
\[ \frac{6}{27} = \frac{2}{9} \]

**Final Answer:**
Thus, the limit is:
\[ \lim_{x \to 27} \frac{(x^{1/3} + 3)(x^{1/3} - 3)}{x - 27} = \frac{2}{9} \]