📚 Topic Summary
The limit of $\frac{sin(x)}{x}$ as $x$ approaches 0 is a fundamental concept in calculus. It doesn't matter if you plug in 0 directly because it results in an indeterminate form $\frac{0}{0}$. L'Hôpital's Rule or geometric arguments are typically employed to prove that: $\lim_{x \to 0} \frac{sin(x)}{x} = 1$. This limit is essential for evaluating many other limits involving trigonometric functions.
🧮 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Limit |
A. A function whose value oscillates between -1 and 1. |
| 2. Indeterminate Form |
B. A value that a function approaches as the input approaches some value. |
| 3. Sine Function |
C. A rule that relates an input to an output. |
| 4. Function |
D. An expression whose value cannot be determined. |
| 5. L'Hôpital's Rule |
E. A technique for evaluating limits of indeterminate forms by taking derivatives. |
(Answers: 1-B, 2-D, 3-A, 4-C, 5-E)
✍️ Part B: Fill in the Blanks
The limit of $\frac{sin(x)}{x}$ as $x$ approaches ____ is equal to ____. This result is crucial in calculus and is often proven using the ______ Theorem or by applying _________ Rule. If you try direct substitution, you'll encounter the __________ form.
(Answers: 0, 1, Squeeze, L'Hôpital's, indeterminate)
🤔 Part C: Critical Thinking
Explain why direct substitution fails when evaluating $\lim_{x \to 0} \frac{sin(x)}{x}$, and outline the steps you would take to correctly determine the limit.