Given that the derivative `f'(a)` exists. Indicate which of the following statements(s) is/are always true?

Options

A. `f'(a)=lim_(htoa)(f(h)-f(a))/(h-a)`
B. `f'(a)=lim_(hto0)(f(a)-f(a-h))/h`
C. `f'(a)=lim_(t to0)(f(a+2t)-f(a))/t`
D. `f'(a)=lim_(t to0)(f(a+2t)-t(a+t))/(2t)`

Answer

To determine which statements are always true given that the derivative \( f'(a) \) exists, we will analyze each option based on the definition of the derivative.

### Step-by-Step Solution:

1. **Understanding the Definition of Derivative**:
The derivative of a function \( f \) at a point \( a \) is defined as:
\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]
This means that for the derivative to exist, this limit must converge to a finite value.

2. **Analyzing Option A**:
The statement is:
\[ f(a) = \lim_{h \to 0} \frac{f(h) - f(a)}{h - a} \]
- Here, we can rewrite this as:
\[ f(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
- This is indeed the definition of the derivative at \( a \) when \( h \) approaches \( 0 \). Thus, **Option A is true**.

3. **Analyzing Option B**:
The statement is:
\[ \lim_{h \to 0} \frac{f(a) - f(a - h)}{h} \]
- We can rewrite this as:
\[ \lim_{h \to 0} \frac{f(a) - f(a - h)}{-h} \]
- This is equivalent to:
\[ -\lim_{h \to 0} \frac{f(a - h) - f(a)}{h} \]
- This limit represents the derivative from the left side at \( a \), which exists if \( f'(a) \) exists. Thus, **Option B is true**.

4. **Analyzing Option C**:
The statement is:
\[ \lim_{t \to 0} \frac{a + 2t - f(a)}{t} \]
- Here, we notice that the expression does not follow the standard form of the derivative, as it involves \( a + 2t \) instead of approaching \( a \). Thus, **Option C is false**.

5. **Analyzing Option D**:
The statement is:
\[ \lim_{t \to 0} \frac{a + 2t - a - t}{t} \]
- Simplifying this gives:
\[ \lim_{t \to 0} \frac{t}{t} = 1 \]
- This does not relate to the derivative of \( f \) at \( a \) and does not follow the definition. Thus, **Option D is false**.

### Conclusion:
The statements that are always true given that the derivative \( f'(a) \) exists are:
- **Option A**: True
- **Option B**: True
- **Option C**: False
- **Option D**: False