Given `g(x)=(1/x), h(x)=x^(2)+2x+(lamda+1)` and `u(x)=1/x+cos(1/(x^(2)))`
Let `f(x)=lim_(ntooo)(x^(2n+1)g(x)+h(x))/(x^(2n)+3x.u(x))`
If `lim_(xto0)f(x)=2`, then the value of `lamda` is

Options

A. 10
B. 5
C. 2
D. Does not exist

Answer

To solve the problem, we need to evaluate the limit of the function \( f(x) \) as \( x \) approaches 0 and find the value of \( \lambda \) such that this limit equals 2.

### Step-by-Step Solution:

1. **Define the Functions**:
We have:
\[ g(x) = \frac{1}{x}, \quad h(x) = x^2 + 2x + (\lambda + 1), \quad u(x) = \frac{1}{x} + \cos\left(\frac{1}{x^2}\right) \]

2. **Write the Expression for \( f(x) \)**:
The function \( f(x) \) is given by:
\[ f(x) = \lim_{n \to \infty} \frac{x^{2n+1} g(x) + h(x)}{x^{2n} + 3x u(x)} \]

3. **Substituting \( g(x) \) and \( h(x) \)**:
Substitute \( g(x) \) and \( h(x) \) into the expression:
\[ f(x) = \lim_{n \to \infty} \frac{x^{2n+1} \cdot \frac{1}{x} + (x^2 + 2x + \lambda + 1)}{x^{2n} + 3x \left(\frac{1}{x} + \cos\left(\frac{1}{x^2}\right)\right)} \]
Simplifying this gives:
\[ f(x) = \lim_{n \to \infty} \frac{x^{2n} + x^2 + 2x + \lambda + 1}{x^{2n} + 3 + 3x \cos\left(\frac{1}{x^2}\right)} \]

4. **Evaluate the Limit as \( n \to \infty \)**:
As \( n \to \infty \), \( x^{2n} \) dominates the numerator and denominator if \( x \neq 0 \):
\[ f(x) = \lim_{n \to \infty} \frac{x^{2n}}{x^{2n}} = 1 \quad \text{(if \( x \neq 0 \))} \]
However, we need to analyze the behavior as \( x \to 0 \).

5. **Taking the Limit as \( x \to 0 \)**:
As \( x \to 0 \):
- The term \( x^{2n} \) approaches 0.
- The numerator simplifies to \( 0 + 0 + 0 + \lambda + 1 = \lambda + 1 \).
- The denominator simplifies to \( 0 + 3 + 0 = 3 \).

Thus, we have:
\[ f(0) = \frac{\lambda + 1}{3} \]

6. **Setting the Limit Equal to 2**:
We are given that:
\[ \lim_{x \to 0} f(x) = 2 \]
Therefore:
\[ \frac{\lambda + 1}{3} = 2 \]

7. **Solving for \( \lambda \)**:
Multiply both sides by 3:
\[ \lambda + 1 = 6 \]
Subtract 1 from both sides:
\[ \lambda = 5 \]

### Conclusion:
The value of \( \lambda \) is \( 5 \).