https://file.aiquickdraw.com/mathbot/file/1725414483325y1br7lyy.jpg

i dont get how the last one is the answer

The image you've provided shows a limit expression: 

\[
\lim_{{x \to 4}} \frac{{f(x) - f(4)}}{{x - 4}}
\]

This expression is indeed used to calculate something important in calculus. Let's break it down:

1. **Average Rate of Change**: This is usually calculated over an interval and gives us the "average" slope between two points. It’s similar to finding the slope of a secant line between two points on a curve.

2. **Instantaneous Rate of Change**: This refers to the slope of the tangent line to the curve at a single point, which is essentially the derivative of the function at that point. The limit process \(\lim_{{x \to 4}}\) is precisely how the derivative is defined. 

### Why is the Limit Expression Calculating the Instantaneous Rate of Change?

The limit expression you're dealing with is the **definition of the derivative** at a point. Specifically:

\[
f'(4) = \lim_{{x \to 4}} \frac{{f(x) - f(4)}}{{x - 4}}
\]

This gives you the instantaneous rate of change of the function \(f(x)\) at \(x = 4\). In other words, it's the slope of the tangent line to the curve at \(x = 4\). 

- The **average rate of change** over an interval from \(x\) to 4 would be simply calculated as \(\frac{{f(x) - f(4)}}{{x - 4}}\) without taking the limit as \(x \to 4\).

- However, the **instantaneous rate of change** at \(x = 4\) requires us to consider what happens as \(x\) gets arbitrarily close to 4, which is precisely what the limit does.

### Conclusion:

The expression \(\lim_{{x \to 4}} \frac{{f(x) - f(4)}}{{x - 4}}\) calculates the instantaneous rate of change at \(x = 4\), also known as the derivative of \(f(x)\) at that point.

Would you like more details or have any further questions?

Here are some related questions for further exploration:

1. What are the differences between a secant line and a tangent line in calculus?
2. How do you calculate the average rate of change over a specific interval?
3. What are some practical examples of instantaneous rates of change in real life?
4. Can the limit expression be used for non-polynomial functions? How?
5. What happens if the limit does not exist? What does it mean in terms of the function's behavior?

**Tip:** Understanding the concept of limits is crucial for mastering calculus. Try visualizing the process of a point getting infinitely close to another to grasp the idea of the derivative better.

Explore how the limit expression lim(x→4) [f(x) - f(4)] / (x - 4) is used to calculate the instantaneous rate of change at x = 4. This explanation covers key calculus concepts, including limits, derivatives, and the definition of the derivative. Suitable for students in Grades 11-12.

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