Solved examples: evaluating 0/0 limits with factoring and cancellation

📚 Quick Study Guide

🔍 When evaluating limits of the form $\frac{0}{0}$, try factoring the numerator and denominator to see if any common factors can be cancelled. 💡 If a common factor $(x - a)$ exists in both the numerator and denominator, cancelling it out can eliminate the indeterminate form and allow you to directly substitute $x = a$. 📝 Remember that factoring techniques like difference of squares ($a^2 - b^2 = (a+b)(a-b)$) and perfect square trinomials ($a^2 + 2ab + b^2 = (a+b)^2$) are often helpful. ➗ After cancelling common factors, re-evaluate the limit by substituting the value of $x$. If the limit now yields a finite number, you've successfully evaluated it. 📈 Limits that can be evaluated using this method are usually continuous at that point once the common factors causing the 0/0 form are removed.

Practice Quiz

1. What is the first step when evaluating a limit that results in the indeterminate form $\frac{0}{0}$?
1. Substitute the value directly.
2. Factor the numerator and denominator.
3. Rationalize the denominator.
4. Use L'Hôpital's Rule.
2. Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.
1. 0
2. 2
3. 4
4. Does not exist
3. Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}$.
1. 6
2. -6
3. $\infty$
4. 0
4. Evaluate $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$.
1. 1
2. 2
3. 3
4. 0
5. What is the factored form of $x^2 - 4x + 4$?
1. $(x - 1)(x - 4)$
2. $(x + 2)(x - 2)$
3. $(x - 2)^2$
4. $(x + 4)(x - 1)$
6. Evaluate $\lim_{x \to -1} \frac{x^2 + 2x + 1}{x + 1}$.
1. -2
2. 0
3. 2
4. 1
7. Evaluate $\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 6x + 5}$.
1. $\frac{5}{2}$
2. $\frac{2}{3}$
3. $\frac{5}{3}$
4. $\frac{3}{2}$