For a certain value of 'c' `lim _(x to oo) [(x ^(5) +7x^(4)+2 )^(c ) -x]` is finite and non-zero. Then the value of limit is :

Options

A. `7/5`
B. `1`
C. `2/5`
D. None of these

Answer

To solve the limit problem, we need to analyze the expression given:

\[ \lim_{x \to \infty} \left[(x^5 + 7x^4 + 2)^c - x\right] \]

### Step 1: Factor out the dominant term
The dominant term in the expression \(x^5 + 7x^4 + 2\) as \(x\) approaches infinity is \(x^5\). We can factor \(x^5\) out of the expression inside the limit:

\[ = \lim_{x \to \infty} \left[x^5 \left(1 + \frac{7}{x} + \frac{2}{x^5}\right)^c - x\right] \]

### Step 2: Simplify the expression
Now, we can simplify the expression further. The limit can be rewritten as:

\[ = \lim_{x \to \infty} \left[x^5 \left(1 + \frac{7}{x} + \frac{2}{x^5}\right)^c - x\right] \]

### Step 3: Analyze the term \(\left(1 + \frac{7}{x} + \frac{2}{x^5}\right)^c\)
As \(x\) approaches infinity, \(\frac{7}{x}\) and \(\frac{2}{x^5}\) approach 0. Thus, we can use the binomial expansion for small values:

\[ \left(1 + \frac{7}{x} + \frac{2}{x^5}\right)^c \approx 1 + c\left(\frac{7}{x} + \frac{2}{x^5}\right) \]

### Step 4: Substitute back into the limit
Substituting this approximation back into the limit gives:

\[ = \lim_{x \to \infty} \left[x^5 \left(1 + c\left(\frac{7}{x} + \frac{2}{x^5}\right)\right) - x\right] \]

Expanding this, we get:

\[ = \lim_{x \to \infty} \left[x^5 + 7cx^4 + 2cx^0 - x\right] \]

### Step 5: Combine terms
Now we can combine the terms:

\[ = \lim_{x \to \infty} \left[x^5 - x + 7cx^4 + 2c\right] \]

### Step 6: Set the limit to be finite and non-zero
For the limit to be finite and non-zero, the highest power of \(x\) must cancel out. This means we need the coefficient of \(x^5\) to be zero:

\[ 1 + 7c = 0 \]

### Step 7: Solve for \(c\)
From the equation \(1 + 7c = 0\), we can solve for \(c\):

\[ 7c = -1 \implies c = -\frac{1}{7} \]

### Step 8: Substitute \(c\) back into the limit
Now, substituting \(c = -\frac{1}{7}\) back into the limit expression, we can find the limit:

\[ \lim_{x \to \infty} \left[-\frac{1}{7}x^5 + 7\left(-\frac{1}{7}\right)x^4 + 2\left(-\frac{1}{7}\right)\right] \]

This simplifies to:

\[ = \lim_{x \to \infty} \left[-\frac{1}{7}x^5 - x^4 - \frac{2}{7}\right] \]

### Step 9: Determine the finite limit
As \(x\) approaches infinity, the dominant term is \(-\frac{1}{7}x^5\), which goes to \(-\infty\). Thus, we need to ensure that the limit is finite and non-zero, which leads us to conclude that the value of \(c\) must be adjusted to ensure the terms balance out.

### Final Answer
The value of \(c\) that makes the limit finite and non-zero is:

\[ c = \frac{1}{5} \]