The value of the limit `lim_(xto0) (a^(sqrt(x))-a^(1//sqrt(x)))/(a^(sqrt(x))+a^(1//sqrt(x))),agt1,`is

Options

A. does not exist
B. `1//3`
C. 0
D. `2//9`

Explanation

To solve the limit

\[ \lim_{x \to 0} \frac{a^{\sqrt{x}} - a^{\frac{1}{\sqrt{x}}}}{a^{\sqrt{x}} + a^{\frac{1}{\sqrt{x}}}} \]

where \( a > 1 \), we will follow these steps:

### Step 1: Analyze the limit as \( x \to 0 \)

As \( x \to 0 \):
- \( \sqrt{x} \to 0 \)
- \( \frac{1}{\sqrt{x}} \to \infty \)

Thus, we can evaluate the terms:
- \( a^{\sqrt{x}} \to a^0 = 1 \)
- \( a^{\frac{1}{\sqrt{x}}} \to a^{\infty} = \infty \)

### Step 2: Substitute the limits into the expression

Substituting these values into the limit gives us:

\[ \frac{1 - \infty}{1 + \infty} = \frac{-\infty}{\infty} \]

This is an indeterminate form of type \( \frac{-\infty}{\infty} \).

### Step 3: Simplify the expression

To resolve the indeterminate form, we can factor out the dominant term \( a^{\frac{1}{\sqrt{x}}} \) from both the numerator and the denominator:

\[ \frac{a^{\frac{1}{\sqrt{x}}} \left( a^{\sqrt{x} - \frac{1}{\sqrt{x}}} - 1 \right)}{a^{\frac{1}{\sqrt{x}}} \left( a^{\sqrt{x}} + 1 \right)} \]

This simplifies to:

\[ \frac{a^{\sqrt{x} - \frac{1}{\sqrt{x}}} - 1}{1 + a^{-\frac{1}{\sqrt{x}}}} \]

### Step 4: Evaluate the limit of the simplified expression

As \( x \to 0 \):
- \( \sqrt{x} - \frac{1}{\sqrt{x}} \to 0 - \infty = -\infty \)
- Thus, \( a^{\sqrt{x} - \frac{1}{\sqrt{x}}} \to a^{-\infty} = 0 \)

So the numerator becomes:

\[ 0 - 1 = -1 \]

For the denominator, as \( x \to 0 \):

\[ 1 + a^{-\frac{1}{\sqrt{x}}} \to 1 + 0 = 1 \]

### Step 5: Final limit evaluation

Thus, we have:

\[ \lim_{x \to 0} \frac{-1}{1} = -1 \]

### Conclusion

The value of the limit is:

\[ \boxed{-1} \]