If `f(x)=x(sqrt(x)-sqrt(x+1))` then f(x) is:

Options

A. A) Continuous but not differentiable at x = 0
B. B) Differentiable at x = 0
C. C) Not continuous at x = 0
D. D) None of the above

Answer

To determine whether the function \( f(x) = x(\sqrt{x} - \sqrt{x+1}) \) is continuous or differentiable at the point \( x = 0 \), we will follow these steps:

### Step 1: Evaluate \( f(0) \)
First, we need to find the value of the function at \( x = 0 \):
\[ f(0) = 0(\sqrt{0} - \sqrt{0+1}) = 0(0 - 1) = 0 \]

### Step 2: Find the Left-Hand Limit as \( x \) Approaches 0
Next, we calculate the left-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x(\sqrt{x} - \sqrt{x+1}) \]
Substituting \( x \) with \( -h \) (where \( h \) is a small positive number):
\[ \lim_{h \to 0^+} (-h)(\sqrt{-h} - \sqrt{-h + 1}) \]
Since \( \sqrt{-h} \) is not defined for real numbers, we can conclude that the left-hand limit does not exist.

### Step 3: Find the Right-Hand Limit as \( x \) Approaches 0
Now, we calculate the right-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x(\sqrt{x} - \sqrt{x+1}) \]
Substituting \( x \) with \( h \):
\[ \lim_{h \to 0^+} h(\sqrt{h} - \sqrt{h+1}) \]
This can be simplified:
\[ \sqrt{h} - \sqrt{h+1} = \frac{h - (h + 1)}{\sqrt{h} + \sqrt{h+1}} = \frac{-1}{\sqrt{h} + \sqrt{h+1}} \]
Thus,
\[ \lim_{h \to 0^+} h \left( \frac{-1}{\sqrt{h} + \sqrt{h+1}} \right) = \lim_{h \to 0^+} \frac{-h}{\sqrt{h} + \sqrt{h+1}} = \frac{-0}{\sqrt{0} + \sqrt{1}} = 0 \]

### Step 4: Compare the Limits and the Function Value
Now we compare the left-hand limit, right-hand limit, and the function value at \( x = 0 \):
- Left-hand limit: Does not exist
- Right-hand limit: 0
- Function value: \( f(0) = 0 \)

Since the left-hand limit does not exist, the function is not continuous at \( x = 0 \).

### Conclusion
Since \( f(x) \) is not continuous at \( x = 0 \), it cannot be differentiable at that point.

Thus, the answer is:
**The function \( f(x) \) is discontinuous at \( x = 0 \).**