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"Calculus 1 - Limits Worksheet 4: Evaluating Limits by Factoring, Part 2" from MathTutorDVD.com. However, it does
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Since there are no specific problems visible in the image, I cannot solve a particular limit problem for you. But I can help explain how to
evaluate limits by factoring, which is the topic of this worksheet.
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🔍 How to Evaluate Limits by Factoring (Overview)
This technique is used when direct substitution into a rational function (a fraction with polynomials) results in an indeterminate form like $ \frac{0}{0} $. In such cases, factoring the numerator and denominator may allow you to cancel common factors and simplify the expression before taking the limit.
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✔ Step-by-Step Strategy:
1.
Try Direct Substitution
- Plug in the value that $ x $ approaches.
- If you get $ \frac{0}{0} $, you have an indeterminate form → proceed to factoring.
2.
Factor Both Numerator and Denominator
- Use techniques like:
- Difference of squares: $ a^2 - b^2 = (a - b)(a + b) $
- Trinomial factoring
- Grouping
- Factoring out common terms
3.
Cancel Common Factors
- Cancel any common factors in the numerator and denominator.
-
Important: This is only valid because we're taking a limit as $ x $ approaches a value, not evaluating at that exact value.
4.
Re-evaluate the Limit
- Substitute the value into the simplified expression.
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📌 Example Problem (Illustrative):
Evaluate:
$$
\lim_{x \to 2} \frac{x^2 - 4}{x - 2}
$$
#### Step 1: Try direct substitution
$$
\frac{2^2 - 4}{2 - 2} = \frac{0}{0} \quad \text{(Indeterminate)}
$$
#### Step 2: Factor
$$
\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}
$$
#### Step 3: Cancel
$$
= x + 2 \quad \text{(for } x \ne 2\text{)}
$$
#### Step 4: Take the limit
$$
\lim_{x \to 2} (x + 2) = 2 + 2 = 4
$$
✔ Final Answer: $ \boxed{4} $
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Parent Tip: Review the logic above to help your child master the concept of evaluating limits worksheet.
