4.7 Using L'Hôpital's Rule

• What is L'Hôpital's Rule used to evaluate?

  Limits of indeterminate forms
• L'Hôpital's Rule states that if $\lim_{x \to a} f(x) =$$0$ and $\lim_{x \to a} g(x) =$$0$, then $\lim_{x \to a} \frac{f(x)}{g(x)} =$$\lim_{x \to a} \frac{f'(x)}{g'(x)}$ is called the new
• L'Hôpital's Rule can only be applied to indeterminate forms.
• For L'Hôpital's Rule to apply, both $f(x)$ and $g(x)$ must be differentiable
• What is the result of applying L'Hôpital's Rule to $\lim_{x \to 0} \frac{\sin(x)}{x}$?


  1
• What are indeterminate forms in mathematics?

  Expressions with no defined value
• Match the indeterminate form with its condition:

  $0 / 0$ ↔️ Both numerator and denominator approach 0

  $\infty / \infty$ ↔️ Both numerator and denominator approach infinity
• L'Hôpital's Rule is always required to resolve indeterminate forms.

  False
• One condition for L'Hôpital's Rule is that both $f(x)$ and $g(x)$ must be differentiable near a
• Steps to apply L'Hôpital's Rule

  1️⃣ Define L'Hôpital's Rule

  2️⃣ Identify indeterminate forms

  3️⃣ Check conditions for use

  4️⃣ Differentiate numerator and denominator separately

  5️⃣ Evaluate the new limit
• The indeterminate form $0 / 0$ occurs when both the numerator and denominator approach 0.
• Both the numerator and denominator grow infinitely large (positive or negative) as $x$ approaches a value a
• Indeterminate forms arise when direct substitution in a limit results in $0 / 0$ or $\infty / \infty$.
• What are the two common indeterminate forms?

  $0 / 0$ and $\infty / \infty$
• The indeterminate form $0 / 0$ occurs when both the numerator and denominator approach 0
• Give an example of a limit that results in the $0 / 0$ indeterminate form.


  \lim_{x \to 0} \frac{\sin(x)}{x}</latex>
• The indeterminate form $\infty / \infty$ occurs when both the numerator and denominator grow infinitely large
• Give an example of a limit that results in the $\infty / \infty$ indeterminate form.


  $\lim_{x \to \infty} \frac{x^{2}}{e^{x}}$
• Steps to apply L'Hôpital's Rule

  1️⃣ Differentiate the numerator and denominator separately

  2️⃣ Apply the derivatives to the limit

  3️⃣ Simplify the new limit

  4️⃣ Evaluate the simplified limit
• Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using L'Hôpital's Rule.


  $1$
• What are the two indeterminate forms for limits that L'Hôpital's Rule helps to evaluate?

  $0 / 0$ and $\infty / \infty$
• For L'Hôpital's Rule to be applied, $f(x)$ andg(x)</latex> must be differentiable
• Steps to apply L'Hôpital's Rule

  1️⃣ Differentiate the numerator and denominator separately

  2️⃣ Apply the derivatives to the limit

  3️⃣ Simplify the new limit and evaluate
• To evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using L'Hôpital's Rule, the derivative of $\sin(x)$ is $\cos(x)$
• What is the value of \lim_{x \to 0} \cos(x)</latex>?

  1
• L'Hôpital's Rule states that if $\lim_{x \to a} f(x) =$$0$ and $\lim_{x \to a} g(x) =$$0$, then $\lim_{x \to a} \frac{f(x)}{g(x)} =$$\lim_{x \to a} \frac{f'(x)}{g'(x)}$True
• Match the indeterminate form with its description:

  $0 / 0$ ↔️ Both numerator and denominator approach 0

  $\infty / \infty$ ↔️ Both numerator and denominator approach infinity
• What are the two conditions for applying L'Hôpital's Rule?

  Indeterminate form and differentiability
• Steps to apply L'Hôpital's Rule

  1️⃣ Differentiate the numerator and denominator separately

  2️⃣ Apply the derivatives to the limit

  3️⃣ Simplify the new limit and evaluate
• After applying L'Hôpital's Rule, we evaluate the limit of the new quotient
• What should you do if the new limit after applying L'Hôpital's Rule is still an indeterminate form?

  Reapply L'Hôpital's Rule
• The value of $\lim_{x \to 0} \frac{e^{x} - 1 - x}{x^{2}}$ using L'Hôpital's Rule is $1 / 2$
• What is the result of applying L'Hôpital's Rule to $\lim_{x \to 0} \frac{e^{x}}{2}$?


  $1 / 2$
• If the new limit after applying L'Hôpital's Rule is still indeterminate, the rule can be applied again.
• What is the initial indeterminate form of $\lim_{x \to 0} \frac{e^{x} - 1 - x}{x^{2}}$?


  $0 / 0$
• After applying L'Hôpital's Rule once to $\lim_{x \to 0} \frac{e^{x} - 1}{2x}$, the new form is still indeterminate.
• Steps for evaluating $\lim_{x \to 0} \frac{e^{x} - 1 - x}{x^{2}}$ using L'Hôpital's Rule


  1️⃣ Check initial conditions

  2️⃣ Apply L'Hôpital's Rule

  3️⃣ Recheck conditions

  4️⃣ Reapply L'Hôpital's Rule

  5️⃣ Evaluate the limit
• What are the two conditions required for reapplying L'Hôpital's Rule?

  Indeterminate form, differentiability
• L'Hôpital's Rule can only be reapplied if the functions are differentiable near the point of interest.
• What are the derivatives of $x^{2}$ and $1 - \cos(x)$ after applying L'Hôpital's Rule?


  $2x$ and $\sin(x)$