The value of
`lim_(xto oo)(2x^(1//2)+3x^(1//3)+4x^(1//4)+....nx^(1//n))/((2x-3)^(1//2)+(2x-3)^(1//3)+....+(2x-3)^(1//n))`is

Options

A. `sqrt(2)`
B. 2
C. `(1)/(sqrt(3))`
D. 0

Answer

To solve the limit

\[ \lim_{x \to \infty} \frac{2x^{1/2} + 3x^{1/3} + 4x^{1/4} + \ldots + nx^{1/n}}{(2x - 3)^{1/2} + (2x - 3)^{1/3} + \ldots + (2x - 3)^{1/n}}, \]

we will analyze both the numerator and the denominator as \(x\) approaches infinity.

### Step 1: Analyze the Numerator

The numerator is

\[ 2x^{1/2} + 3x^{1/3} + 4x^{1/4} + \ldots + nx^{1/n}. \]

As \(x\) approaches infinity, the term with the highest power will dominate. The highest power in the numerator is \(nx^{1/n}\). Therefore, we can factor out \(x^{1/n}\):

\[ = x^{1/n} \left( 2x^{1/2 - 1/n} + 3x^{1/3 - 1/n} + 4x^{1/4 - 1/n} + \ldots + n \right). \]

### Step 2: Analyze the Denominator

The denominator is

\[ (2x - 3)^{1/2} + (2x - 3)^{1/3} + \ldots + (2x - 3)^{1/n}. \]

As \(x\) approaches infinity, we can approximate \(2x - 3 \approx 2x\). Thus, we can factor out \(x^{1/2}\):

\[ = x^{1/2} \left( (2 - \frac{3}{x})^{1/2} + (2 - \frac{3}{x})^{1/3} + \ldots + (2 - \frac{3}{x})^{1/n} \right). \]

### Step 3: Simplifying the Limit

Now, substituting these factorizations back into the limit gives us:

\[ \lim_{x \to \infty} \frac{x^{1/n} \left( 2x^{1/2 - 1/n} + 3x^{1/3 - 1/n} + \ldots + n \right)}{x^{1/2} \left( (2 - \frac{3}{x})^{1/2} + (2 - \frac{3}{x})^{1/3} + \ldots + (2 - \frac{3}{x})^{1/n} \right)}. \]

### Step 4: Canceling Common Factors

We can cancel \(x^{1/n}\) from the numerator and \(x^{1/2}\) from the denominator:

\[ = \lim_{x \to \infty} \frac{2x^{1/2 - 1/n} + 3x^{1/3 - 1/n} + \ldots + n}{(2 - \frac{3}{x})^{1/2} + (2 - \frac{3}{x})^{1/3} + \ldots + (2 - \frac{3}{x})^{1/n}}. \]

### Step 5: Evaluating the Limit

As \(x\) approaches infinity, \(\frac{3}{x} \to 0\), and thus the denominator approaches:

\[ (2)^{1/2} + (2)^{1/3} + \ldots + (2)^{1/n}. \]

The numerator approaches \(2\) since the term \(2x^{1/2 - 1/n}\) will dominate.

### Step 6: Final Calculation

Thus, we have:

\[ \lim_{x \to \infty} \frac{2}{(2)^{1/2}} = \frac{2}{\sqrt{2}} = \sqrt{2}. \]

### Conclusion

The final answer is:

\[ \sqrt{2}. \]