Скачать или смотреть AP Calculus AB 1.2 Evaluating Limits Algebraically Involving Multiplication (Example 3)

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*How to Calculate a Limit Algebraically:*

1. *Direct Substitution:*
The first step when finding a limit algebraically is to substitute the value of \( x \) that the limit approaches directly into the function.
Example:
\[
\lim_{x \to 3} (x^2 - 1)
\]
Substituting \( x = 3 \) gives:
\[
3^2 - 1 = 9 - 1 = 8
\]
Therefore, \( \lim_{x \to 3} (x^2 - 1) = 8 \).

2. *Simplify the Expression:*
If direct substitution results in an indeterminate form (like \( \frac{0}{0} \)), try simplifying the expression using factoring, canceling terms, or other algebraic techniques.
Example:
\[
\lim_{x \to 2} \frac{x^2 - 4}{x - 2}
\]
Factor the numerator:
\[
\lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}
\]
Cancel \( (x - 2) \):
\[
\lim_{x \to 2} (x + 2) = 2 + 2 = 4
\]
Therefore, \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4 \).

3. *Rationalizing (for square roots):*
If the function involves square roots, you can multiply the numerator and denominator by the conjugate to eliminate the square root.
Example:
\[
\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}
\]
Multiply by the conjugate:
\[
\frac{\sqrt{x} - 2}{x - 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2} = \frac{(\sqrt{x})^2 - 2^2}{(x - 4)(\sqrt{x} + 2)} = \frac{x - 4}{(x - 4)(\sqrt{x} + 2)}
\]
Cancel \( (x - 4) \):
\[
\lim_{x \to 4} \frac{1}{\sqrt{x} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Therefore, \( \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{4} \).

By following these steps, you can solve most limits algebraically without the need for advanced techniques like L'Hopital's Rule or considering piecewise functions.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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