If ` lim_( xto 2 ) (f(x) -f(2))/( x-2) ` exist ,then

Options

A. ` underset(xto2^(-) )lim f(x) ne underset(xto2^(+) ) lim f(x) `
B. ` f(x)` is differentiable
C. ` underset (xto2) lim f(x) =f(2) `
D. `underset(xto 2 )lim f(x) ne f(2) `

Answer

To solve the problem, we need to analyze the given limit:

\[ \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2} \]

### Step 1: Understanding the Limit
The limit given in the problem is a standard form of the definition of the derivative of the function \( f(x) \) at the point \( x = 2 \).

### Step 2: Recognizing the Derivative
From the definition of the derivative, we know that:

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

In our case, \( a = 2 \). Therefore, if the limit exists, it implies that:

\[ f'(2) = \lim_{x \to 2} \frac{f(x) - f(2)}{x - 2} \]

### Step 3: Conclusion about Differentiability
Since the limit exists, it indicates that the function \( f(x) \) is differentiable at \( x = 2 \). Differentiability at a point implies that the function is continuous at that point.

### Step 4: Implications of Differentiability
If \( f(x) \) is differentiable at \( x = 2 \), it means:

1. The left-hand limit as \( x \) approaches 2 from the left equals the right-hand limit as \( x \) approaches 2 from the right.
2. The function is continuous at \( x = 2 \).

### Final Conclusion
Thus, we conclude that if the limit exists, then \( f(x) \) is differentiable at \( x = 2 \).