Let `lim_(x to 1) (x^(4)-1)/(x-1)=lim_(k to k ) (x^(3)-k^(3))/(x^(2)-k^(2))` then value of k is

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(4)/(3)`
D. `(8)/(3)`

Answer

To solve the given limit problem, we will follow these steps:

### Step 1: Evaluate the left-hand limit
We need to evaluate the limit:
\[ \lim_{x \to 1} \frac{x^4 - 1}{x - 1} \]
First, we notice that substituting \(x = 1\) gives us the indeterminate form \(0/0\). Therefore, we can factor the numerator.

### Step 2: Factor the numerator
The expression \(x^4 - 1\) can be factored using the difference of squares:
\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 1} \frac{(x - 1)(x + 1)(x^2 + 1)}{x - 1} \]
We can cancel \(x - 1\) from the numerator and denominator (as long as \(x \neq 1\)):
\[ \lim_{x \to 1} (x + 1)(x^2 + 1) \]

### Step 3: Substitute \(x = 1\)
Now, we can substitute \(x = 1\):
\[ (1 + 1)(1^2 + 1) = 2 \cdot 2 = 4 \]
Thus, we have:
\[ \lim_{x \to 1} \frac{x^4 - 1}{x - 1} = 4 \]

### Step 4: Evaluate the right-hand limit
Now we need to evaluate the limit:
\[ \lim_{x \to k} \frac{x^3 - k^3}{x^2 - k^2} \]
Similar to the previous limit, substituting \(x = k\) gives us the indeterminate form \(0/0\). We can factor both the numerator and the denominator.

### Step 5: Factor the numerator and denominator
The expression \(x^3 - k^3\) can be factored as:
\[ x^3 - k^3 = (x - k)(x^2 + kx + k^2) \]
And the expression \(x^2 - k^2\) can be factored as:
\[ x^2 - k^2 = (x - k)(x + k) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to k} \frac{(x - k)(x^2 + kx + k^2)}{(x - k)(x + k)} \]
We can cancel \(x - k\) from the numerator and denominator (as long as \(x \neq k\)):
\[ \lim_{x \to k} \frac{x^2 + kx + k^2}{x + k} \]

### Step 6: Substitute \(x = k\)
Now we can substitute \(x = k\):
\[ \frac{k^2 + k^2 + k^2}{k + k} = \frac{3k^2}{2k} = \frac{3k}{2} \]

### Step 7: Set the limits equal
Since both limits must be equal, we have:
\[ 4 = \frac{3k}{2} \]

### Step 8: Solve for \(k\)
To find \(k\), we multiply both sides by 2:
\[ 8 = 3k \]
Now, divide both sides by 3:
\[ k = \frac{8}{3} \]

Thus, the value of \(k\) is:
\[ \boxed{\frac{8}{3}} \]