If function `f(x) = (sqrt(1+x) - root(3)(1+x))/(x)` is continuous function at x = 0, then f(0) is equal to

Options

A. 2
B. `(1)/(4)`
C. `(1)/(6)`
D. `(1)/(3)`

Answer

To find \( f(0) \) for the function

\[ f(x) = \frac{\sqrt{1+x} - \sqrt[3]{1+x}}{x} \]

and ensure that it is continuous at \( x = 0 \), we need to evaluate the limit as \( x \) approaches 0.

### Step 1: Check the limit
We need to compute

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt[3]{1+x}}{x} \]

### Step 2: Substitute \( x = 0 \)
Substituting \( x = 0 \) directly into the function gives:

\[ f(0) = \frac{\sqrt{1+0} - \sqrt[3]{1+0}}{0} = \frac{1 - 1}{0} = \frac{0}{0} \]

This is an indeterminate form, so we need to apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule
L'Hôpital's Rule states that if we have an indeterminate form \( \frac{0}{0} \), we can take the derivatives of the numerator and denominator:

1. The derivative of the numerator \( \sqrt{1+x} - \sqrt[3]{1+x} \):
- The derivative of \( \sqrt{1+x} \) is \( \frac{1}{2\sqrt{1+x}} \).
- The derivative of \( \sqrt[3]{1+x} \) is \( \frac{1}{3(1+x)^{2/3}} \).

Thus, the derivative of the numerator is:

\[ \frac{1}{2\sqrt{1+x}} - \frac{1}{3(1+x)^{2/3}} \]

2. The derivative of the denominator \( x \) is simply \( 1 \).

Now we can rewrite the limit:

\[ \lim_{x \to 0} \frac{\frac{1}{2\sqrt{1+x}} - \frac{1}{3(1+x)^{2/3}}}{1} \]

### Step 4: Evaluate the limit
Now we substitute \( x = 0 \):

\[ \frac{1}{2\sqrt{1+0}} - \frac{1}{3(1+0)^{2/3}} = \frac{1}{2 \cdot 1} - \frac{1}{3 \cdot 1} = \frac{1}{2} - \frac{1}{3} \]

### Step 5: Find a common denominator
To subtract these fractions, we find a common denominator:

\[ \frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{1}{3} = \frac{2}{6} \]

Thus,

\[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

### Conclusion
Therefore,

\[ \lim_{x \to 0} f(x) = \frac{1}{6} \]

Since \( f(x) \) is continuous at \( x = 0 \), we have:

\[ f(0) = \frac{1}{6} \]

### Final Answer
\[ f(0) = \frac{1}{6} \]

---