Evaluating limits by direct substitution is a fundamental technique in calculus. It involves plugging in the value that $x$ approaches directly into the function. If the result is a real number, then that number is the limit. However, if you get an indeterminate form (like $\frac{0}{0}$), you'll need to use other methods such as factoring, rationalizing, or L'Hôpital's Rule.
This technique works well for continuous functions. A continuous function is one where you can draw its graph without lifting your pen from the paper. Polynomials, exponentials, and trigonometric functions (within their domains) are often continuous, making direct substitution a handy tool for evaluating their limits.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Limit | A. A function whose limit exists at a point. |
| 2. Direct Substitution | B. The value that a function approaches as the input approaches some value. |
| 3. Continuous Function | C. A method to evaluate limits by plugging in the value directly. |
| 4. Indeterminate Form | D. An expression whose value cannot be determined by direct substitution, such as $\frac{0}{0}$. |
| 5. L'Hôpital's Rule | E. A rule that can be used to evaluate limits of indeterminate forms. |
Evaluating limits by _____ substitution involves plugging in the value that $x$ approaches directly into the _____. If the result is a _____ number, then that number is the _____. However, if you get an _____ form, you'll need to use other methods.
Explain why direct substitution doesn't work when evaluating $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ directly. What algebraic technique can you use to find the limit?
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