https://file.aiquickdraw.com/mathbot/file/1726575513703bq1eu3h9.jpg

Let's address the two problems in the image:

### Problem 1:  
*Bukitkan fungsi \( f \) terdiferensial di titik \( x = x_0 \) jika dan hanya jika:*

\[
\lim_{x \to x_0} \frac{f(x) - f(x_0) - m(x - x_0)}{x - x_0} = 0
\]

This is a condition for differentiability at \( x_0 \) where \( m \) is constant, and it implies that:

\[
f'(x_0) = m
\]

### Solution:
This limit is essentially the definition of the derivative at \( x_0 \), but adjusted to include a linear approximation (i.e., the tangent line with slope \( m \)).

1. **Rewrite the expression for differentiability**:  
   The standard definition of the derivative is:

   \[
   f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}
   \]

   Now, the given equation adds a linear term \( m(x - x_0) \) and states that for \( f \) to be differentiable at \( x_0 \), the difference between \( f(x) \) and this linear approximation should vanish faster than \( (x - x_0) \).

2. **Interpret the equation**:  
   The equation:

   \[
   \lim_{x \to x_0} \frac{f(x) - f(x_0) - m(x - x_0)}{x - x_0} = 0
   \]

   suggests that \( f(x) \) is approximated near \( x_0 \) by the line \( f(x_0) + m(x - x_0) \), meaning the derivative of \( f \) at \( x_0 \) is exactly \( m \). Hence, this condition implies \( f'(x_0) = m \), since the additional terms vanish as \( x \to x_0 \).

### Problem 2:
*Bukitkan: Jika \( f \) terdefinisi pada persekitaran titik \( c \), maka \( f \) terdiferensial di titik \( c \) jika dan hanya jika titik diskontinu \( c \) dari fungsi*

\[
h(x) = \frac{f(x) - f(c)}{x - c}
\]

### Solution:
This problem essentially asks you to prove that \( f \) is differentiable at \( c \) if and only if the function \( h(x) = \frac{f(x) - f(c)}{x - c} \) has no discontinuity at \( c \).

1. **Continuity of \( h(x) \)**:  
   The function \( h(x) \) is the difference quotient, which is the foundation of the derivative. For \( f \) to be differentiable at \( c \), the limit:

   \[
   \lim_{x \to c} h(x) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}
   \]

   must exist and be finite. This is exactly the definition of \( f'(c) \).

2. **Discontinuity at \( c \)**:  
   If \( f \) is not differentiable at \( c \), then this limit will either not exist or will be infinite, implying that \( h(x) \) is discontinuous at \( c \). Conversely, if \( f \) is differentiable, then \( h(x) \) will be continuous at \( c \).

---

Would you like any further details or clarifications on these problems?

Here are 5 follow-up questions that relate to this discussion:

1. What is the geometrical interpretation of differentiability at a point?
2. How does the existence of the limit in Problem 1 guarantee that \( f'(x_0) = m \)?
3. Can a function be continuous but not differentiable? Provide an example.
4. What happens if the function \( f \) has a cusp at \( x_0 \)? Will the condition in Problem 1 still hold?
5. How does the difference quotient in Problem 2 relate to the derivative definition?

**Tip**: When proving differentiability, always check for both the existence of the limit and continuity at the point in question.

1. Bukitkan fungsi f terdiferensial di titik x = x0 jika dan hanya jika limit (x -> x0) dari (f(x) - f(x0) - m(x - x0)) / (x - x0) = 0 untuk m konstan. Dalam kasus ini f'(x0) = m.
2. Bukitkan: Jika f terdefinisi pada persekitaran titik c, maka f terdiferensial di titik c jika dan hanya jika titik diskontinu c dari fungsi h(x) = (f(x) - f(c)) / (x - c) dapat dirubah.

This page explains how to prove differentiability at a point using limits and the difference quotient. Learn how to prove f'(x0) = m for constant m and check differentiability at a discontinuous point using the function h(x). Suitable for advanced calculus students.

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