`lim_(xtooo) ((2x+1)^(40)(4x+1)^(5))/((2x+3)^(45))` is equal to

Options

A. `(1)/(2pi)`
B. `(-1)/(pi)`
C. `(-2)/(pi)`
D. none of these

Explanation

To solve the limit \( \lim_{x \to \infty} \frac{(2x+1)^{40}(4x+1)^{5}}{(2x+3)^{45}} \), we will follow these steps:

### Step 1: Identify the form of the limit
As \( x \) approaches infinity, both the numerator and denominator approach infinity. Thus, we have an indeterminate form of type \( \frac{\infty}{\infty} \).

**Hint:** Always check the behavior of the function as \( x \) approaches the limit to determine if it results in an indeterminate form.

### Step 2: Factor out the highest power of \( x \)
We can factor out \( x \) from each term in the numerator and denominator:

\[ (2x + 1)^{40} = x^{40} \left(2 + \frac{1}{x}\right)^{40} \]
\[ (4x + 1)^{5} = x^{5} \left(4 + \frac{1}{x}\right)^{5} \]
\[ (2x + 3)^{45} = x^{45} \left(2 + \frac{3}{x}\right)^{45} \]

### Step 3: Rewrite the limit
Substituting these factored forms into the limit gives:

\[ \lim_{x \to \infty} \frac{x^{40} \left(2 + \frac{1}{x}\right)^{40} \cdot x^{5} \left(4 + \frac{1}{x}\right)^{5}}{x^{45} \left(2 + \frac{3}{x}\right)^{45}} \]

### Step 4: Simplify the expression
The \( x^{40} \) and \( x^{5} \) in the numerator combine to \( x^{45} \), which cancels with \( x^{45} \) in the denominator:

\[ \lim_{x \to \infty} \frac{\left(2 + \frac{1}{x}\right)^{40} \cdot \left(4 + \frac{1}{x}\right)^{5}}{\left(2 + \frac{3}{x}\right)^{45}} \]

### Step 5: Evaluate the limit
As \( x \to \infty \), \( \frac{1}{x} \to 0 \). Therefore, we can substitute:

\[ \left(2 + \frac{1}{x}\right) \to 2, \quad \left(4 + \frac{1}{x}\right) \to 4, \quad \text{and} \quad \left(2 + \frac{3}{x}\right) \to 2 \]

This gives us:

\[ \lim_{x \to \infty} \frac{(2)^{40} \cdot (4)^{5}}{(2)^{45}} \]

### Step 6: Simplify the final expression
Calculating the powers:

\[ = \frac{2^{40} \cdot 4^{5}}{2^{45}} = \frac{2^{40} \cdot (2^2)^{5}}{2^{45}} = \frac{2^{40} \cdot 2^{10}}{2^{45}} = \frac{2^{50}}{2^{45}} = 2^{5} = 32 \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to \infty} \frac{(2x+1)^{40}(4x+1)^{5}}{(2x+3)^{45}} = 32 \]