Detailed Examples of Simplifying 0/0 Indeterminate Limits in Calculus

📚 Quick Study Guide

🔍 L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the limit exists.

💡 Factoring: Try factoring both the numerator and the denominator to see if any common factors cancel out.

➕ Conjugate Multiplication: If dealing with square roots, multiply both the numerator and the denominator by the conjugate of the expression containing the square root.

📐 Trigonometric Identities: Use trigonometric identities (e.g., $\sin^2(x) + \cos^2(x) = 1$) to simplify the expression.

🪜 Algebraic Manipulation: Rearrange or rewrite the expression to eliminate the indeterminate form.

Practice Quiz

1. What is the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1?
1. 0
2. 1
3. 2
4. Does not exist


2. Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$.
1. 0
2. 1
3. $\infty$
4. -1


3. Find the limit of $\frac{x^2 - 4}{x - 2}$ as $x$ approaches 2.
1. 0
2. 2
3. 4
4. Does not exist


4. What is $\lim_{x \to 0} \frac{e^x - 1}{x}$?
1. 0
2. 1
3. $\infty$
4. -1


5. Determine $\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}$.
1. 0
2. 1/2
3. 1
4. $\infty$


6. Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.
1. 0
2. 3
3. 6
4. Does not exist


7. What is $\lim_{x \to 0} \frac{\tan(x)}{x}$?
1. 0
2. 1
3. $\infty$
4. -1