Question

If \(x\) is so large that terms containing \(x^{-3}\), \(x^{-4}\), \(x^{-5}\), \ldots can be neglected, then the approximate value of \[ \left(\frac{3x - 5}{4x^2 + 3}\right)^{-4/5} \] is:

• A. \(\left(\frac{3}{4x}\right)^{4/5} \left(1 - \frac{4}{3x} - \frac{7}{5x^2}\right)\)
• B. \(\left(\frac{4x}{3}\right)^{4/5} \left(1 + \frac{4}{3x} + \frac{13}{5x^2}\right)\)
• C. \(\left(\frac{4x}{3}\right)^{4/5} \left(1 + \frac{4}{3x} - \frac{13}{5x^2}\right)\)
• D. \(\left(\frac{3}{4x}\right)^{4/5} \left(1 - \frac{4}{3x} + \frac{7}{5x^2}\right)\)

Hint:

Use binomial expansion to approximate expressions with large variables and neglect higher powers.

Answer 1

The Correct Option is B

Step 1: Rewrite the expression
\[ \left(\frac{3x - 5}{4x^2 + 3}\right)^{-4/5} = \left(\frac{4x^2 + 3}{3x - 5}\right)^{4/5} \] Step 2: Approximate numerator and denominator for large \(x\)
\[ 4x^2 + 3 \approx 4x^2 \left(1 + \frac{3}{4x^2}\right), \quad 3x - 5 \approx 3x \left(1 - \frac{5}{3x}\right) \] Step 3: Express as product
\[ \left(\frac{4x^2}{3x}\right)^{4/5} \left(\frac{1 + \frac{3}{4x^2}}{1 - \frac{5}{3x}}\right)^{4/5} = \left(\frac{4x}{3}\right)^{4/5} \left(1 + \frac{3}{4x^2}\right)^{4/5} \left(1 - \frac{5}{3x}\right)^{-4/5} \] Step 4: Use binomial expansion
\[ (1 + u)^r \approx 1 + r u + \frac{r(r-1)}{2} u^2 \] Apply to both factors with: \[ u_1 = \frac{3}{4x^2}, \quad r = \frac{4}{5} \] \[ u_2 = -\frac{5}{3x}, \quad r = -\frac{4}{5} \] Calculate expansions and multiply. Step 5: Result
\[ \left(\frac{4x}{3}\right)^{4/5} \left(1 + \frac{4}{3x} + \frac{13}{5x^2}\right) \]