`lim_(x->oo)(1+a^2)^xsinb/((1+a^2)^x),` where `a!=0` (a) Both f (x) and g(x) are non-differentiable at 5 points (b) `a^(2)b` (c) `a^(2)//b` (d) `b`

Options

A. Both f (x) and g(x) are non-differentiable at 5 points
B. `a^(2)b`
C. `a^(2)//b`
D. `b`

Answer

To solve the limit problem

\[ \lim_{x \to \infty} \frac{(1 + a^2)^x \sin b}{(1 + a^2)^x} \]

where \( a \neq 0 \), we can simplify the expression step by step.

### Step 1: Simplify the expression
The limit can be simplified as follows:

\[ \lim_{x \to \infty} \frac{(1 + a^2)^x \sin b}{(1 + a^2)^x} = \lim_{x \to \infty} \sin b \]

### Step 2: Evaluate the limit
Since \(\sin b\) is a constant with respect to \(x\), the limit simplifies to:

\[ \sin b \]

### Final Result
Thus, the limit evaluates to:

\[ \sin b \]

### Conclusion
The answer to the limit is \( \sin b \).