Let `f(x)=(3)/(4)x+1,f^(n)(x)` be defined as `f^(2)(x)=f(f(x)),` and for `n ge 2, f^(n+1)(x)=f(f^(n)(x))." If " lambda =lim_(n to oo) f^(n)(x),` then

Options

A. `lambda` is independent of x
B. `lambda` is a linear polynomial in x
C. the line `y=lambda` has slope 0
D. the line `4y=lambda` touches the unit circle with center at the origin.

Explanation

To solve the problem, we need to analyze the function \( f(x) = \frac{3}{4}x + 1 \) and find the limit \( \lambda = \lim_{n \to \infty} f^{(n)}(x) \).

### Step-by-step Solution:

1. **Define the Function**:
We start with the function:
\[ f(x) = \frac{3}{4}x + 1 \]

2. **Calculate \( f^2(x) \)**:
To find \( f^2(x) \), we apply \( f \) to itself:
\[ f^2(x) = f(f(x)) = f\left(\frac{3}{4}x + 1\right) \]
Substituting into \( f \):
\[ f^2(x) = \frac{3}{4}\left(\frac{3}{4}x + 1\right) + 1 = \frac{3}{4} \cdot \frac{3}{4}x + \frac{3}{4} + 1 = \frac{9}{16}x + \frac{3}{4} + 1 \]
Simplifying:
\[ f^2(x) = \frac{9}{16}x + \frac{7}{4} \]

3. **Calculate \( f^3(x) \)**:
Now we find \( f^3(x) = f(f^2(x)) \):
\[ f^3(x) = f\left(\frac{9}{16}x + \frac{7}{4}\right) = \frac{3}{4}\left(\frac{9}{16}x + \frac{7}{4}\right) + 1 \]
Simplifying:
\[ f^3(x) = \frac{27}{64}x + \frac{21}{16} + 1 = \frac{27}{64}x + \frac{37}{16} \]

4. **Generalizing \( f^n(x) \)**:
We can observe a pattern. The \( n \)-th iteration seems to follow:
\[ f^{(n)}(x) = \left(\frac{3}{4}\right)^n x + C_n \]
where \( C_n \) is a constant that we need to determine.

5. **Finding \( C_n \)**:
From our calculations, we can see that:
\[ C_n = \frac{3}{4}C_{n-1} + 1 \]
This is a recursive relation. If we solve this, we can find \( C_n \) explicitly.

6. **Finding the Limit**:
As \( n \to \infty \), \( \left(\frac{3}{4}\right)^n \to 0 \). Therefore, we need to find \( \lim_{n \to \infty} C_n \).

Solving the recursion:
\[ C_n = \frac{3}{4}C_{n-1} + 1 \]
Assuming \( C_n \) approaches a limit \( L \):
\[ L = \frac{3}{4}L + 1 \implies L - \frac{3}{4}L = 1 \implies \frac{1}{4}L = 1 \implies L = 4 \]

7. **Final Result**:
Thus, we have:
\[ \lambda = \lim_{n \to \infty} f^{(n)}(x) = 0 + 4 = 4 \]

### Conclusion:
The limit \( \lambda \) is:
\[ \lambda = 4 \]