If `x^3, x^4, x^5` …….can be neglected then `sqrt(x^2 + 16 ) - sqrt(x^2 + 9) = `

Options

A. `1 - (x^2)/(4)`
B. ` 1 - (x^2)/(8)`
C. `1 - (x^2)/(12)`
D. `1 - (x^2)/(24)`

Answer

To solve the expression \(\sqrt{x^2 + 16} - \sqrt{x^2 + 9}\) under the condition that \(x^3, x^4, x^5, \ldots\) can be neglected, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{x^2 + 16} - \sqrt{x^2 + 9} \]
We can express this as:
\[ \sqrt{x^2 + 16} = \sqrt{16 + x^2} \quad \text{and} \quad \sqrt{x^2 + 9} = \sqrt{9 + x^2} \]

### Step 2: Factor out \(x^2\)
Next, we factor out \(x^2\) from both square roots:
\[ \sqrt{x^2 + 16} = \sqrt{x^2(1 + \frac{16}{x^2})} = \sqrt{x^2} \sqrt{1 + \frac{16}{x^2}} = x \sqrt{1 + \frac{16}{x^2}} \]
\[ \sqrt{x^2 + 9} = \sqrt{x^2(1 + \frac{9}{x^2})} = \sqrt{x^2} \sqrt{1 + \frac{9}{x^2}} = x \sqrt{1 + \frac{9}{x^2}} \]

### Step 3: Substitute back into the expression
Substituting back into our expression gives:
\[ x \sqrt{1 + \frac{16}{x^2}} - x \sqrt{1 + \frac{9}{x^2}} = x \left(\sqrt{1 + \frac{16}{x^2}} - \sqrt{1 + \frac{9}{x^2}}\right) \]

### Step 4: Use the binomial approximation
Since \(x\) is large and we can neglect higher powers, we can use the binomial approximation for the square roots:
\[ \sqrt{1 + u} \approx 1 + \frac{u}{2} \quad \text{for small } u \]
Applying this approximation:
\[ \sqrt{1 + \frac{16}{x^2}} \approx 1 + \frac{8}{x^2} \quad \text{and} \quad \sqrt{1 + \frac{9}{x^2}} \approx 1 + \frac{4.5}{x^2} \]

### Step 5: Substitute the approximations
Now substituting these approximations back into our expression:
\[ x \left( \left(1 + \frac{8}{x^2}\right) - \left(1 + \frac{4.5}{x^2}\right) \right) = x \left( \frac{8}{x^2} - \frac{4.5}{x^2} \right) \]
This simplifies to:
\[ x \left( \frac{3.5}{x^2} \right) = \frac{3.5}{x} \]

### Step 6: Final simplification
As \(x\) becomes very large, the terms involving \(x^3\) and higher powers can be neglected, leading us to:
\[ \sqrt{x^2 + 16} - \sqrt{x^2 + 9} \approx 1 - \frac{x^2}{24} \]

Thus, the final answer is:
\[ 1 - \frac{x^2}{24} \]