If ` underset( x to a ) lim (x^(5) - a^(5))/( x-a) = 405` , then possible values ( s ) of a:

Options

A. `1`
B. `-1`
C. `+-3`
D. `0 `

Answer

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

### Step 1: Set up the limit expression
We start with the limit expression:
\[ \lim_{x \to a} \frac{x^5 - a^5}{x - a} = 405 \]

### Step 2: Identify the indeterminate form
If we substitute \(x = a\) directly into the expression, we get:
\[ \frac{a^5 - a^5}{a - a} = \frac{0}{0} \]
This is an indeterminate form, which means we can apply L'Hospital's Rule.

### Step 3: Apply L'Hospital's Rule
According to L'Hospital's Rule, we differentiate the numerator and denominator:
- The derivative of the numerator \(x^5 - a^5\) with respect to \(x\) is \(5x^4\).
- The derivative of the denominator \(x - a\) with respect to \(x\) is \(1\).

Thus, we rewrite the limit as:
\[ \lim_{x \to a} \frac{5x^4}{1} \]

### Step 4: Evaluate the limit
Now we can substitute \(x = a\) into the simplified expression:
\[ 5a^4 = 405 \]

### Step 5: Solve for \(a^4\)
To find \(a^4\), we divide both sides by 5:
\[ a^4 = \frac{405}{5} = 81 \]

### Step 6: Solve for \(a\)
Now we take the fourth root of both sides:
\[ a = \pm 3 \]

### Conclusion
The possible values of \(a\) are:
\[ a = 3 \quad \text{or} \quad a = -3 \]

Thus, the correct answer is option (C) \( \pm 3 \).

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