Direct substitution limits test questions for high school calculus students

📚 Quick Study Guide

• 🔢 Direct substitution is the simplest method for evaluating limits.
• ✏️ If $f(x)$ is a polynomial, rational, trigonometric, exponential, or logarithmic function, and $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.
• ⚠️ Be careful with rational functions! Check that the denominator is not zero at $x = a$. If it is, direct substitution won't work.
• 💡 When direct substitution results in an indeterminate form like $\frac{0}{0}$, try algebraic manipulation first (factoring, rationalizing, etc.).
• 📝 Common functions where direct substitution usually works: polynomials, $sin(x)$, $cos(x)$, $e^x$.

🧪 Practice Quiz

1. What is $\lim_{x \to 2} (x^2 + 3x - 1)$?
   1. 5
   2. 9
   3. 10
   4. 11
2. What is $\lim_{x \to 0} cos(x)$?
   1. 0
   2. 1
   3. -1
   4. $\infty$
3. What is $\lim_{x \to 1} \frac{x^2 + 1}{x + 1}$?
   1. 0
   2. 1
   3. 2
   4. $\frac{1}{2}$
4. What is $\lim_{x \to -1} (5x^3 - 2x + 3)$?
   1. 0
   2. -4
   3. 10
   4. -10
5. What is $\lim_{x \to \pi} sin(x)$?
   1. 0
   2. 1
   3. -1
   4. $\pi$
6. What is $\lim_{x \to 3} e^x$?
   1. 0
   2. 1
   3. $e^3$
   4. $\infty$
7. What is $\lim_{x \to 4} \sqrt{x + 5}$?
   1. 1
   2. 2
   3. 3
   4. 4