`lim_(x->a){[(a^(1/2)+x^(1/2))/(a^(1/4)-x^(1/4)))^(- 1)-(2(a x)^(1/4))/(x^(3/4)-a^(1/4)x^(1/2)+a^(1/2)x^(1/4)-a^(3/4))]^(- 1)-sqrt2^(log_4 a)}^8`

Options

A. a
B. `a^(3//4)`
C. `a^(2)`
D. None of these

Answer

To solve the limit problem step by step, we start with the given expression:

\[ \lim_{x \to a} \left[ \left( \frac{a^{1/2} + x^{1/2}}{a^{1/4} - x^{1/4}} \right)^{-1} - \frac{2(a x)^{1/4}}{x^{3/4} - a^{1/4} x^{1/2} + a^{1/2} x^{1/4} - a^{3/4}} \right]^{-1} - \sqrt{2}^{\log_4 a})^8 \]

### Step 1: Rewrite the expression

We start by rewriting the limit expression for clarity:

\[ \lim_{x \to a} \left[ \frac{a^{1/4} - x^{1/4}}{a^{1/2} + x^{1/2}} - \frac{2(a x)^{1/4}}{x^{3/4} - a^{1/4} x^{1/2} + a^{1/2} x^{1/4} - a^{3/4}} \right]^{-1} - \sqrt{2}^{\log_4 a} \]

### Step 2: Simplify the first term

Next, we simplify the first term:

\[ \frac{a^{1/4} - x^{1/4}}{a^{1/2} + x^{1/2}} = \frac{(a^{1/4} - x^{1/4})}{(a^{1/2} + x^{1/2})} \]

As \(x\) approaches \(a\), both the numerator and denominator approach \(0\), leading us to apply L'Hôpital's Rule or factorization.

### Step 3: Apply L'Hôpital's Rule

Using L'Hôpital's Rule, we differentiate the numerator and denominator:

- The derivative of the numerator \(a^{1/4} - x^{1/4}\) is \(-\frac{1}{4} x^{-3/4}\).
- The derivative of the denominator \(a^{1/2} + x^{1/2}\) is \(\frac{1}{2} x^{-1/2}\).

Thus, we have:

\[ \lim_{x \to a} \frac{-\frac{1}{4} x^{-3/4}}{\frac{1}{2} x^{-1/2}} = \lim_{x \to a} \frac{-\frac{1}{4}}{\frac{1}{2} x^{1/4}} = -\frac{1}{2} \cdot \frac{1}{4^{1/4}} = -\frac{1}{2 \sqrt[4]{4}} = -\frac{1}{4} \]

### Step 4: Simplify the second term

Next, we simplify the second term:

\[ \frac{2(a x)^{1/4}}{x^{3/4} - a^{1/4} x^{1/2} + a^{1/2} x^{1/4} - a^{3/4}} \]

As \(x\) approaches \(a\), we again find that both the numerator and denominator approach \(0\). We can apply L'Hôpital's Rule again or factor.

### Step 5: Evaluate the limit

After applying L'Hôpital's Rule or simplifying, we find that the limit approaches \(0\). Thus, we have:

\[ \lim_{x \to a} \left[ \text{First term} - \text{Second term} \right] = 0 - 0 = 0 \]

### Step 6: Final expression

Now substituting back into the original limit expression, we find:

\[ \lim_{x \to a} \left[ 0 - \sqrt{2}^{\log_4 a} \right] = -\sqrt{2}^{\log_4 a} \]

### Step 7: Evaluate \(\sqrt{2}^{\log_4 a}\)

Using properties of logarithms:

\[ \sqrt{2}^{\log_4 a} = 2^{1/2 \cdot \log_4 a} = 2^{\log_2 a / 4} = a^{1/4} \]

### Step 8: Final result

Thus, we have:

\[ \lim_{x \to a} \left[ -a^{1/4} \right]^8 = -a^2 \]

### Conclusion

The final answer is:

\[ -a^2 \]