Test questions for evaluating limits using direct substitution

📚 Quick Study Guide

• 🔍 Direct substitution is a method to evaluate limits by plugging in the value that $x$ approaches into the function.
• 💡 If $f(x)$ is a polynomial or rational function and $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.
• 📝 For rational functions, be careful of division by zero. If direct substitution results in $\frac{0}{0}$ or $\frac{c}{0}$ (where $c$ is a non-zero constant), other methods like factoring, rationalizing, or L'Hôpital's Rule are required.
• ➕ Constant Rule: $\lim_{x \to a} c = c$
• ✖️ Constant Multiple Rule: $\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x)$
• ➗ Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided that $\lim_{x \to a} g(x) \neq 0$
• 🌱 Power Rule: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

🧪 Practice Quiz

1. What is the value of $\lim_{x \to 2} (x^2 + 3x - 1)$?
1. 5
2. 9
3. 10
4. 13
2. Evaluate $\lim_{x \to -1} (2x^3 - x + 5)$.
1. 2
2. 4
3. 6
4. 8
3. Find the limit: $\lim_{x \to 0} \frac{x^2 + 2x}{x + 1}$.
1. 0
2. 1
3. 2
4. Does Not Exist
4. What is the limit of $\lim_{x \to 3} \sqrt{x^2 + 7}$?
1. 2
2. 4
3. 6
4. 8
5. Determine the limit $\lim_{x \to 4} (5x - 7)$.
1. 10
2. 11
3. 12
4. 13
6. Calculate $\lim_{x \to -2} \frac{x^2 - 4}{x + 2}$.
1. -4
2. -2
3. 0
4. 4
7. Compute $\lim_{x \to 1} (x^4 - 3x^2 + 2)$.
1. 0
2. 1
3. 2
4. 3