`lim_(x to infty) a^(x) sin (b/a^(x))`, where `a gt 1` =

Options

A. b log a
B. a log b
C. b
D. a

Answer

To solve the limit \( \lim_{x \to \infty} a^x \sin\left(\frac{b}{a^x}\right) \) where \( a > 1 \), we can follow these steps:

### Step 1: Rewrite the Limit Expression
We start with the limit:
\[ L = \lim_{x \to \infty} a^x \sin\left(\frac{b}{a^x}\right) \]

### Step 2: Analyze the Argument of the Sine Function
As \( x \to \infty \), \( a^x \) grows very large since \( a > 1 \). Consequently, \( \frac{b}{a^x} \) approaches 0. Thus, we can rewrite the sine function:
\[ \sin\left(\frac{b}{a^x}\right) \approx \frac{b}{a^x} \quad \text{(using the small angle approximation, } \sin z \approx z \text{ as } z \to 0\text{)} \]

### Step 3: Substitute the Approximation into the Limit
Substituting this approximation back into our limit gives:
\[ L = \lim_{x \to \infty} a^x \cdot \frac{b}{a^x} = \lim_{x \to \infty} b = b \]

### Step 4: Conclude the Limit
Thus, we find that:
\[ L = b \]

### Final Answer
Therefore, the limit is:
\[ \lim_{x \to \infty} a^x \sin\left(\frac{b}{a^x}\right) = b \]

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