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What is an indeterminate form?
An expression whose limit cannot be evaluated directly, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
What does L'Hôpital's Rule help evaluate?
Limits of indeterminate forms.
What is L'Hôpital's Rule?
If $\lim_{x\to a}\frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$.
How to evaluate $\lim_{x\to a}\frac{f(x)}{g(x)}$ using L'Hôpital's Rule?
1. Check if the limit is in indeterminate form. 2. Verify conditions: $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$ or both approach $\pm \infty$. 3. Apply L'Hôpital's Rule: $\lim_{x\to a}\frac{f'(x)}{g'(x)}$. 4. Evaluate the new limit.
Steps to solve $\lim_{x \to 0} \frac{\sin(3x)}{x}$ using L'Hopital's Rule.
1. Check indeterminate form: $\frac{0}{0}$. 2. Apply L'Hopital's Rule: $\lim_{x \to 0} \frac{3\cos(3x)}{1}$. 3. Evaluate: $3\cos(0) = 3$.
How to evaluate $\lim_{x\to \infty} \frac{3x^2 - 8}{7x^2 + 21}$?
1. Check indeterminate form: $\frac{\infty}{\infty}$. 2. Apply L'Hôpital's Rule: $\lim_{x\to \infty} \frac{6x}{14x}$. 3. Simplify: $\frac{6}{14} = \frac{3}{7}$.