Let f(x) = { x ×e^1/x  -1(- 1 here is not in power) / e^1/x  +1(+1 here is not in power) if x not equal to 0 

0 if x=0

Examine the derivability at x=0

To examine the derivability of the function \( f(x) \) at \( x = 0 \), we need to check if the derivative \( f'(0) \) exists. 

The function \( f(x) \) is defined as:

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

To check the derivability at \( x = 0 \), we need to determine if the limit:

\[ 
f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}
\]

exists. Since \( f(0) = 0 \), this simplifies to:

\[ 
f'(0) = \lim_{h \to 0} \frac{f(h)}{h}
\]

Let's examine this limit:

1. For \( x \neq 0 \):

\[ 
f(x) = \frac{x e^{1/x} - 1}{e^{1/x} + 1}
\]

2. We need to compute \( \lim_{h \to 0} \frac{f(h)}{h} \):

\[ 
\lim_{h \to 0} \frac{1}{h} \left( \frac{h e^{1/h} - 1}{e^{1/h} + 1} \right)
\]

We can simplify this expression further. 

First, consider the numerator and denominator separately:
\[ h e^{1/h} - 1 \quad \text{and} \quad e^{1/h} + 1 \]

As \( h \to 0 \), \( e^{1/h} \) tends to \( \infty \).

So,

\[ 
\lim_{h \to 0} \frac{h e^{1/h} - 1}{h (e^{1/h} + 1)} 
\]

Dividing numerator and denominator by \( e^{1/h} \), we get:

\[ 
\lim_{h \to 0} \frac{\frac{h e^{1/h}}{e^{1/h}} - \frac{1}{e^{1/h}}}{\frac{h (e^{1/h} + 1)}{e^{1/h}}} = \lim_{h \to 0} \frac{h - \frac{1}{e^{1/h}}}{h (1 + \frac{1}{e^{1/h}})} 
\]

Since \( \frac{1}{e^{1/h}} \to 0 \) as \( h \to 0 \):

\[ 
\lim_{h \to 0} \frac{h - 0}{h (1 + 0)} = \lim_{h \to 0} \frac{h}{h} = 1
\]

Thus,

\[ 
f'(0) = 1 
\]

Therefore, the derivative of \( f(x) \) at \( x = 0 \) exists and is equal to 1. The function \( f(x) \) is derivable at \( x = 0 \).

---

Would you like further details or have any questions? Here are some questions you might consider:

1. How can I verify the limit process for other functions?
2. What is the significance of differentiability at a point?
3. How can I determine if a function is continuous at a point?
4. What are the general steps for finding a derivative using limits?
5. How can I apply this method to piecewise functions?

**Tip:** Always check the continuity of a function before exploring its differentiability at a given point.

Explore the derivability of the function f(x) at x = 0 using limit definitions and piecewise function properties. Understand how to determine if a function is differentiable at a specific point. Suitable for advanced high school students studying calculus.

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