Yes, you're correct that in the instantaneous rate of change formula:

$\frac{f(x) - f(a)}{x - a}$

as $x$ gets closer and closer to $a$, the distance $x - a$ becomes smaller and smaller.

Why the limit is as $x \to a$:

1. Instantaneous rate of change: This formula represents the average rate of change of the function $f(x)$ over the interval from $a$ to $x$. But the goal is to find the instantaneous rate of change, or the rate at which $f(x)$ is changing exactly at $x = a$.

2. Approaching a single point: To get the instantaneous rate of change, we need to shrink the interval between $x$ and $a$ to essentially zero. As $x$ moves closer and closer to $a$, the change in $x$ ($x - a$) becomes smaller and smaller, which refines the average rate of change into a precise measurement at $x = a$.

3. Limit process: By taking the limit of the difference quotient $\frac{f(x) - f(a)}{x - a}$ as $x \to a$, we essentially zoom in on the behavior of the function at that exact point. The limit allows us to mathematically capture the instantaneous rate of change at $a$, which is the slope of the tangent line to the curve at $x = a$.

4. Why not at $x = a$ directly?: If we plug in $x = a$ directly into the formula, we get a division by zero ($\frac{0}{0}$), which is undefined. That's why we use the limit process. The limit tells us the value that the difference quotient approaches as $x$ gets arbitrarily close to $a$, without actually being equal to $a$.

5. Derivative definition: This leads to the formal definition of the derivative, which is the instantaneous rate of change of $f(x)$ at $a$:

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

The limit allows us to transition from an average rate of change over an interval to the precise rate of change at a single point.

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Would you like more details on any part of this explanation?

Here are 5 related questions to consider:

1. What does the difference quotient represent geometrically on a graph?
2. How does the concept of limits apply to other areas of calculus besides derivatives?
3. How do we know if the limit exists when $x \to a$?
4. What is the relationship between the derivative and the tangent line to a curve?
5. How is the instantaneous rate of change used in real-world applications?

Tip: Always check if the function is continuous at the point $x = a$ when applying the derivative definition!