Let `f(x)={{:((2^((1)/(x))-1)/(2^((1)/(x))+1),":",xne0),(0,":,x=0):}`, then `f(x)` is

Options

A. continuous and differentiable at x = 0
B. continuous but not differentiable at x = 0
C. differentiable but not continuous at x = 0
D. none of these

Answer

To determine the nature of the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} \frac{2^{\frac{1}{x}} - 1}{2^{\frac{1}{x}} + 1} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]

we need to check whether it is continuous and differentiable at \( x = 0 \).

### Step 1: Check Continuity at \( x = 0 \)

To check for continuity at \( x = 0 \), we need to find the left-hand limit (LHL) and right-hand limit (RHL) as \( x \) approaches 0, and see if they equal \( f(0) \).

**Left-Hand Limit (LHL):**

\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{2^{\frac{1}{x}} - 1}{2^{\frac{1}{x}} + 1} \]

As \( x \) approaches \( 0 \) from the left (\( x \to 0^- \)), \( \frac{1}{x} \) approaches \( -\infty \). Thus, \( 2^{\frac{1}{x}} \) approaches \( 0 \).

\[ \text{LHL} = \frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1 \]

**Right-Hand Limit (RHL):**

\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{2^{\frac{1}{x}} - 1}{2^{\frac{1}{x}} + 1} \]

As \( x \) approaches \( 0 \) from the right (\( x \to 0^+ \)), \( \frac{1}{x} \) approaches \( +\infty \). Thus, \( 2^{\frac{1}{x}} \) approaches \( +\infty \).

\[ \text{RHL} = \frac{+\infty - 1}{+\infty + 1} = \frac{+\infty}{+\infty} = 1 \]

### Step 2: Compare Limits and Function Value

Now, we compare the limits with the function value at \( x = 0 \):

\[ f(0) = 0 \]

We have:

- LHL = -1
- RHL = 1
- \( f(0) = 0 \)

Since LHL ≠ RHL and neither of them equals \( f(0) \), we conclude that \( f(x) \) is not continuous at \( x = 0 \).

### Step 3: Check Differentiability at \( x = 0 \)

A function must be continuous at a point to be differentiable there. Since \( f(x) \) is not continuous at \( x = 0 \), it cannot be differentiable at that point.

### Conclusion

Thus, the function \( f(x) \) is:

- Not continuous at \( x = 0 \)
- Not differentiable at \( x = 0 \)

The correct answer is: **None of these.**

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