https://file.aiquickdraw.com/mathbot/file/1729984286578xxkveovu.jpg

To solve this limit problem, let's examine the expression carefully:

\[
\lim_{{h \to 0}} \frac{2(1 + h)^4 - (1 + h)^3 + 2(1 + h)^2 - 3}{h}
\]

We will expand each term in the numerator and simplify before taking the limit.

### Step 1: Expanding Each Term
1. **Expand \((1 + h)^4\):**
   \[
   (1 + h)^4 = 1 + 4h + 6h^2 + 4h^3 + h^4
   \]
   So, \(2(1 + h)^4 = 2 + 8h + 12h^2 + 8h^3 + 2h^4\).

2. **Expand \((1 + h)^3\):**
   \[
   (1 + h)^3 = 1 + 3h + 3h^2 + h^3
   \]
   So, \(-(1 + h)^3 = -1 - 3h - 3h^2 - h^3\).

3. **Expand \((1 + h)^2\):**
   \[
   (1 + h)^2 = 1 + 2h + h^2
   \]
   So, \(2(1 + h)^2 = 2 + 4h + 2h^2\).

### Step 2: Substitute and Simplify
Now, substitute these expanded forms into the expression:

\[
2(1 + h)^4 - (1 + h)^3 + 2(1 + h)^2 - 3 = (2 + 8h + 12h^2 + 8h^3 + 2h^4) - (1 + 3h + 3h^2 + h^3) + (2 + 4h + 2h^2) - 3
\]

Combine like terms:

- Constant terms: \(2 - 1 + 2 - 3 = 0\)
- Terms with \(h\): \(8h - 3h + 4h = 9h\)
- Terms with \(h^2\): \(12h^2 - 3h^2 + 2h^2 = 11h^2\)
- Terms with \(h^3\): \(8h^3 - h^3 = 7h^3\)
- Terms with \(h^4\): \(2h^4\)

So the numerator simplifies to:
\[
9h + 11h^2 + 7h^3 + 2h^4
\]

### Step 3: Factor Out \(h\)
\[
\frac{9h + 11h^2 + 7h^3 + 2h^4}{h} = 9 + 11h + 7h^2 + 2h^3
\]

### Step 4: Take the Limit as \(h \to 0\)
As \(h \to 0\), the terms containing \(h\), \(h^2\), and \(h^3\) will approach zero, leaving:
\[
\lim_{{h \to 0}} \left(9 + 11h + 7h^2 + 2h^3\right) = 9
\]

### Final Answer:
The limit is:
\[
\boxed{9}
\]

This matches option **A**.

Would you like further details or have any questions about any of the steps?

---

Here are 5 related questions:
1. How do you handle higher powers in limits when \( h \) approaches zero?
2. What are other methods to simplify expressions involving limits besides expansion?
3. Why does factoring out \( h \) help in solving limits of this form?
4. How does L'Hôpital's Rule apply to similar limit problems?
5. What is the significance of higher-order terms (like \( h^2, h^3 \)) as \( h \to 0\)?

**Tip:** When dealing with polynomial expansions in limits, always organize and simplify terms carefully to see what can be canceled or made negligible as the variable approaches zero.

Find \( \lim_{{h \to 0}} \frac{2(1 + h)^4 - (1 + h)^3 + 2(1 + h)^2 - 3}{h} \)

Solve the limit problem \( \lim_{{h \to 0}} \frac{2(1 + h)^4 - (1 + h)^3 + 2(1 + h)^2 - 3}{h} \) with step-by-step algebraic expansion and polynomial simplification. This problem is ideal for students in Grades 11-12, focusing on limit laws and binomial expansion.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation