Calculus Quiz Guidance: Limits and Squeeze Theorem

Q1. Evaluate the limit: $$\lim_{h \to 0} \frac{f(-2 + h) - f(-2)}{h}$$ where $$f(x) = 6x^2 + 6$$.

Background

Topic: Definition of the Derivative / Limits

This question tests your understanding of the limit definition of the derivative at a point, specifically using the difference quotient. It also checks your ability to substitute and simplify expressions involving functions.

Key Terms and Formulas:

• Difference Quotient: $$\frac{f(a + h) - f(a)}{h}$$

• Limit: $$\lim_{h \to 0}$$

• Quadratic Function: $$f(x) = 6x^2 + 6$$

Step-by-Step Guidance

1. Start by substituting $a = -2$ into the difference quotient: $$\frac{f(-2 + h) - f(-2)}{h}$$

2. Calculate $f(-2 + h)$: $$f(-2 + h) = 6(-2 + h)^2 + 6$$

3. Expand $(-2 + h)^2$ to get $h^2 - 4h + 4$, then substitute back: $$f(-2 + h) = 6(h^2 - 4h + 4) + 6$$

4. Calculate $f(-2)$: $$f(-2) = 6(-2)^2 + 6 = 6 \cdot 4 + 6 = 24 + 6 = 30$$

5. Set up the difference quotient: $$\frac{[6(h^2 - 4h + 4) + 6] - 30}{h}$$

Try solving on your own before revealing the answer!

Q2. Suppose $$8x - 20 \leq f(x) \leq x^2 + 4x - 16$$. Use this to compute $$\lim_{x \to 2} f(x)$$.

Background

Topic: Squeeze Theorem / Limits

This question is about using the Squeeze Theorem to find the limit of a function that is bounded between two other functions. The Squeeze Theorem is a fundamental concept in calculus for evaluating limits when direct substitution is not possible.

Key Terms and Formulas:

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Limit of a polynomial: Polynomials are continuous, so $\lim_{x \to a} p(x) = p(a)$.

Step-by-Step Guidance

1. Write the inequalities: $$8x - 20 \leq f(x) \leq x^2 + 4x - 16$$

2. Take the limit as $x \to 2$ for each part: $$\lim_{x \to 2} (8x - 20) \leq \lim_{x \to 2} f(x) \leq \lim_{x \to 2} (x^2 + 4x - 16)$$

3. Evaluate the left and right limits using direct substitution: $$8 \cdot 2 - 20$$ and $$2^2 + 4 \cdot 2 - 16$$

4. Simplify both expressions to see if they yield the same value.

Try solving on your own before revealing the answer!