4.7 Using L'Hôpital's Rule

• Match the indeterminate form with its example:

  0/0 ↔️ lim (x→2) (x² - 4) / (x - 2)

  ∞/∞ ↔️ lim (x→∞) (3x² + 2) / (x² + 1)

  0 * ∞ ↔️ lim (x→0) x * ln(x)
• What does L'Hôpital's Rule state?

  f'(x) / g'(x)
• Applying L'Hôpital's Rule to $lim_{x \to 0} \frac{\sin x}{x}$ gives $lim_{x \to 0} \frac{\cos x}{1}$, which equals 1
• Steps to apply L'Hôpital's Rule

  1️⃣ Check for indeterminate form

  2️⃣ Find derivatives of f(x) and g(x)

  3️⃣ Apply L'Hôpital's Rule: $lim_{x \to c} \frac{f'(x)}{g'(x)}$

  4️⃣ Re-evaluate if necessary
• $lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$ is an example of the indeterminate form $0 / 0$.


  True
• What is the purpose of L'Hôpital's Rule?

  Evaluate indeterminate limits
• Which condition must be met for L'Hôpital's Rule to be applied?

  g'(x) ≠ 0
• What is the first step in applying L'Hôpital's Rule?

  Evaluate f'(x) / g'(x)
• Steps to apply L'Hôpital's Rule when the limit is of the form 0 / 0</latex>:

  1️⃣ Check for indeterminate form $0 / 0$

  2️⃣ Find derivatives of $f(x)$ and $g(x)$

  3️⃣ Apply L'Hôpital's Rule: $lim_{x \to c} \frac{f'(x)}{g'(x)}$

  4️⃣ Re-evaluate indeterminate form if necessary
• Match the indeterminate form with its example:

  0/0 ↔️ $lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$

  ∞/∞ ↔️ lim_{x \to \infty} \frac{3x^{2} + 2}{x^{2} + 1}

  0 * ∞ ↔️ $lim_{x \to 0} x \cdot \ln(x)$
• L'Hôpital's Rule requires that $g'(x)$ is not equal to zero near c
• Steps to apply L'Hôpital's Rule to the indeterminate form 0 / 0</latex>:

  1️⃣ Check for indeterminate form $0 / 0$

  2️⃣ Find derivatives of $f(x)$ and $g(x)$

  3️⃣ Apply L'Hôpital's Rule: $lim_{x \to c} \frac{f'(x)}{g'(x)}$
• L'Hôpital's Rule can only be applied if the limit results in an indeterminate form.


  True
• When the limit $lim_{x \to c} \frac{f(x)}{g(x)}$ yields the indeterminate form $0 / 0$, L'Hôpital's Rule can be applied.


  True
• If $\frac{f'(c)}{g'(c)}$ is still an indeterminate form, you must repeat the differentiation process.


  True
• L'Hôpital's Rule can also be applied to limits of the form $\infty / \infty$.


  True
• In the example $lim_{x \to \infty} \frac{6x + 2}{2x}$, applying L'Hôpital's Rule again gives a limit of 3.


  True
• In the example $lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$, applying L'Hôpital's Rule results in a limit of 4
• To apply L'Hôpital's Rule to $\infty / \infty$, you must first confirm that the limit is of the form \infty / \infty
• An indeterminate form means the limit is undefined.

  False
• The indeterminate form $1^{\infty}$ appears in the example lim_{x \to \infty} (1 + 1/x)^x</latex>, which is related to the number e
• L'Hôpital's Rule can only be applied if g'(x) = 0.

  False
• What are the two common indeterminate forms that L'Hôpital's Rule addresses?

  0/0 and ∞/∞
• lim_{x \to \infty} \frac{3x^{2} + 2}{x^{2} + 1} is an example of the indeterminate form $\infty / \infty$, and its limit is 3
• L'Hôpital's Rule works by taking the limit of the original functions.

  False
• Applying L'Hôpital's Rule to $lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$ gives $lim_{x \to 2} \frac{2x}{1}$, which equals 4
• If the ratio of derivatives $\frac{f'(c)}{g'(c)}$ is still an indeterminate form, you must repeat steps 2 and 3 until a determinable limit is achieved
• An indeterminate form in calculus automatically means the limit is undefined.

  False
• What is the purpose of L'Hôpital's Rule?

  Evaluate indeterminate limits
• The example $lim_{x \to 0} \frac{\sin x}{x}$ evaluates to 1 using L'Hôpital's Rule.


  True
• L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as 0 / 0
• When applying L'Hôpital's Rule, you replace the original limit with the limit of the derivatives
• Steps to apply L'Hôpital's Rule to the indeterminate form $0 / 0$


  1️⃣ Check for Indeterminate Form

  2️⃣ Find Derivatives

  3️⃣ Apply L'Hôpital's Rule

  4️⃣ Re-evaluate Indeterminate Form if Needed