If `f(x)={xe^-[1/(|x|)+1/x]; x != 0; 0;x=0` Prove that `f(x)` is not differentiable at `x = 0`

Answer

To prove that the function \( f(x) \) is not differentiable at \( x = 0 \), we will follow these steps:

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} x e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]

### Step 2: Check continuity at \( x = 0 \)
To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0 from both sides.

#### Left-hand limit:
For \( x < 0 \):
\[ f(x) = x e^{-\left(\frac{1}{-x} + \frac{1}{x}\right)} = x e^{-\left(-\frac{1}{x} + \frac{1}{x}\right)} = x e^{0} = x \]
Thus,
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x = 0 \]

#### Right-hand limit:
For \( x > 0 \):
\[ f(x) = x e^{-\left(\frac{1}{x} + \frac{1}{x}\right)} = x e^{-\frac{2}{x}} \]
As \( x \to 0^+ \), \( e^{-\frac{2}{x}} \) approaches 0 much faster than \( x \) approaches 0, hence:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x e^{-\frac{2}{x}} = 0 \]

Since both the left-hand limit and right-hand limit equal \( f(0) = 0 \), we conclude that \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check differentiability at \( x = 0 \)
To check differentiability, we need to find the left-hand derivative and right-hand derivative at \( x = 0 \).

#### Left-hand derivative:
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{h - 0}{h} = \lim_{h \to 0^-} 1 = 1 \]

#### Right-hand derivative:
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h e^{-\frac{2}{h}} - 0}{h} = \lim_{h \to 0^+} e^{-\frac{2}{h}} \]
As \( h \to 0^+ \), \( e^{-\frac{2}{h}} \) approaches 0. Thus,
\[ f'(0^+) = 0 \]

### Step 4: Conclusion
Since the left-hand derivative \( f'(0^-) = 1 \) and the right-hand derivative \( f'(0^+) = 0 \) are not equal, we conclude that \( f(x) \) is not differentiable at \( x = 0 \).

### Final Result
Therefore, we have shown that \( f(x) \) is not differentiable at \( x = 0 \).

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