limit as delta x approaches 0, (x + delta x)^2 - 2(x + delta x) + 1 - (x^2-2x+1) all over delta x

Let's evaluate the limit as \(\Delta x\) approaches 0 of the given expression:

\[
\lim_{\Delta x \to 0} \frac{(x + \Delta x)^2 - 2(x + \Delta x) + 1 - (x^2 - 2x + 1)}{\Delta x}
\]

### Step 1: Expand the expression
First, expand the expression \((x + \Delta x)^2 - 2(x + \Delta x) + 1\):

\[
(x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2
\]

\[
-2(x + \Delta x) = -2x - 2\Delta x
\]

So, the expression becomes:

\[
(x^2 + 2x\Delta x + (\Delta x)^2) - 2x - 2\Delta x + 1
\]

### Step 2: Simplify the expression
Now, subtract the original function \(x^2 - 2x + 1\) from this:

\[
(x^2 + 2x\Delta x + (\Delta x)^2 - 2x - 2\Delta x + 1) - (x^2 - 2x + 1)
\]

Simplifying by canceling out similar terms:

\[
x^2 - x^2 + 2x\Delta x + (\Delta x)^2 - 2x + 2x - 2\Delta x + 1 - 1 = 2x\Delta x + (\Delta x)^2 - 2\Delta x
\]

So the expression becomes:

\[
\frac{2x\Delta x + (\Delta x)^2 - 2\Delta x}{\Delta x}
\]

### Step 3: Factor and simplify further
Factor \(\Delta x\) out of the numerator:

\[
\frac{\Delta x(2x + \Delta x - 2)}{\Delta x}
\]

Cancel out \(\Delta x\) from the numerator and denominator:

\[
2x + \Delta x - 2
\]

### Step 4: Evaluate the limit
Now, take the limit as \(\Delta x \to 0\):

\[
\lim_{\Delta x \to 0} (2x + \Delta x - 2) = 2x - 2
\]

So the final answer is:

\[
\boxed{2x - 2}
\]

Do you want further details or have any questions?

Here are some related questions to explore:
1. How does this limit relate to the derivative of a function?
2. What happens if you change the function slightly, for example by adding a different constant?
3. How would you find the limit if the expression were more complex?
4. Can you find the derivative of \(x^2 - 2x + 1\) directly?
5. How would this process change if we used a different method for differentiation?

**Tip:** Remember that limits are the foundation of derivatives. Understanding how to simplify expressions using limits will make it easier to grasp the concept of differentiation.

limit as delta x approaches 0, (x + delta x)^2 - 2(x + delta x) + 1 - (x^2 - 2x + 1) all over delta x

Explore how to compute the limit as Δx approaches 0 for a quadratic expression involving (x + Δx)^2 - 2(x + Δx) + 1. This problem walks through key calculus concepts such as limits, derivatives, and algebraic simplification, making it ideal for students in Grades 11-12 or introductory calculus courses.

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