`lim_(x to infty) (root(4)(x^(5) + 2) - root(3)(x^(2)+1))/(root(5)(x^(4)+2)-root(2)(x^(3)+1))`=

Options

A. `-1`
B. 0
C. 1
D. `infty`

Answer

To solve the limit

\[ \lim_{x \to \infty} \frac{\sqrt[4]{x^5 + 2} - \sqrt[3]{x^2 + 1}}{\sqrt[5]{x^4 + 2} - \sqrt{ x^3 + 1}} \]

we will follow these steps:

### Step 1: Identify the dominant terms
As \(x\) approaches infinity, the dominant terms in the numerator and denominator will dictate the behavior of the limit.

- In the numerator, \(\sqrt[4]{x^5 + 2}\) behaves like \(\sqrt[4]{x^5}\) and \(\sqrt[3]{x^2 + 1}\) behaves like \(\sqrt[3]{x^2}\).
- In the denominator, \(\sqrt[5]{x^4 + 2}\) behaves like \(\sqrt[5]{x^4}\) and \(\sqrt{x^3 + 1}\) behaves like \(\sqrt{x^3}\).

### Step 2: Rewrite the expression
We can rewrite the limit as:

\[ \lim_{x \to \infty} \frac{\sqrt[4]{x^5} - \sqrt[3]{x^2}}{\sqrt[5]{x^4} - \sqrt{x^3}} \]

### Step 3: Factor out the highest powers
Now, factor out the highest powers from the numerator and denominator:

- From the numerator:
\[ \sqrt[4]{x^5} = x^{5/4} \quad \text{and} \quad \sqrt[3]{x^2} = x^{2/3} \]
Thus, we can factor \(x^{5/4}\):
\[ x^{5/4} \left(1 - \frac{x^{2/3}}{x^{5/4}}\right) = x^{5/4} \left(1 - x^{-1/12}\right) \]

- From the denominator:
\[ \sqrt[5]{x^4} = x^{4/5} \quad \text{and} \quad \sqrt{x^3} = x^{3/2} \]
Thus, we can factor \(x^{4/5}\):
\[ x^{4/5} \left(1 - \frac{x^{3/2}}{x^{4/5}}\right) = x^{4/5} \left(1 - x^{3/10}\right) \]

### Step 4: Substitute back into the limit
Now substituting back into the limit gives us:

\[ \lim_{x \to \infty} \frac{x^{5/4} \left(1 - x^{-1/12}\right)}{x^{4/5} \left(1 - x^{3/10}\right)} \]

### Step 5: Simplify the limit
Now simplify the limit:

\[ = \lim_{x \to \infty} x^{5/4 - 4/5} \cdot \frac{1 - x^{-1/12}}{1 - x^{3/10}} \]

Calculating the exponent:

\[ 5/4 - 4/5 = \frac{25 - 16}{20} = \frac{9}{20} \]

So, we have:

\[ = \lim_{x \to \infty} x^{9/20} \cdot \frac{1 - x^{-1/12}}{1 - x^{3/10}} \]

### Step 6: Analyze the limit
As \(x \to \infty\):

- \(x^{-1/12} \to 0\)
- \(x^{3/10} \to \infty\)

Thus, the limit simplifies to:

\[ = \lim_{x \to \infty} x^{9/20} \cdot \frac{1 - 0}{1 - \infty} = x^{9/20} \cdot \frac{1}{-\infty} \to 0 \]

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

\[ \lim_{x \to \infty} \frac{\sqrt[4]{x^5 + 2} - \sqrt[3]{x^2 + 1}}{\sqrt[5]{x^4 + 2} - \sqrt{x^3 + 1}} = -1 \]