https://file.aiquickdraw.com/mathbot/file/172602101039227yo0u0m.jpg

Let's analyze the mathematical problems in the image.

1. **First problem**:

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

This is a limit problem, where we are asked to compute the derivative of a function using the definition of the derivative.

---

2. **Second problem**:

The problem gives the following conditions:

\[
3x - 2 \leq f(x) \leq x^2 - 3x + 7 \quad \text{for} \quad x \geq 0
\]

And asks for:

\[
\lim_{x \to 3} f(x)
\]

This is a Squeeze Theorem problem where the limit is constrained by two functions.

---

3. **Third problem**:

Given:

\[
8x \leq g(x) \leq 4x^4 - x^2 + 8 \quad \text{for all} \quad x
\]

The problem asks for:

\[
\lim_{x \to 1} g(x)
\]

Again, this seems like another Squeeze Theorem problem. 

---

### Solutions:

1. **First limit problem**: 
We'll solve using the difference quotient for the derivative.

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

- We rewrite the expression in a common denominator and then simplify using algebraic methods. After simplification, this limit gives the derivative of \(\frac{1}{x^2}\), which is \(-\frac{2}{x^3}\).

So, the result is:

\[
\boxed{-\frac{2}{x^3}}
\]

2. **Second limit problem**:
The Squeeze Theorem gives us:

\[
3x - 2 \leq f(x) \leq x^2 - 3x + 7
\]

At \(x = 3\), both the lower and upper bounds give the same result:

\[
f(3) = 7
\]

Thus, by the Squeeze Theorem:

\[
\lim_{x \to 3} f(x) = 7
\]

3. **Third limit problem**:
We apply the Squeeze Theorem again:

\[
8x \leq g(x) \leq 4x^4 - x^2 + 8
\]

For \(x = 1\), both the bounds are equal to \(8\), so:

\[
\lim_{x \to 1} g(x) = 8
\]

Would you like further details or explanation on any of these steps?

---

### Five follow-up questions:

1. How is the definition of a derivative related to the first problem?
2. Can you explain how to solve limits using the Squeeze Theorem?
3. What are some common strategies to simplify algebraic expressions when working with limits?
4. Why is finding a common denominator important in solving the first limit problem?
5. How can you apply the Squeeze Theorem in general for limit problems?

---

### Tip:
When solving limits involving complex fractions, simplifying the numerator or denominator with algebraic identities (like factoring or using common denominators) is a key step to avoid errors.

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE). First problem: lim (h → 0) [(1/(x+h)^2) - (1/x^2)]/h. Second problem: If 3x - 2 ≤ f(x) ≤ x^2 - 3x + 7 for x ≥ 0, find lim (x → 3) f(x). Third problem: If 8x ≤ g(x) ≤ 4x^4 - x^2 + 8 for all x, evaluate lim (x → 1) g(x).

Learn how to solve calculus problems involving limits, derivatives, and the Squeeze Theorem. Includes detailed solutions for evaluating limits, finding derivatives using limit definitions, and applying the Squeeze Theorem. Suitable for Grades 11-12 or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation