4.7 Using L'Hôpital's Rule

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

  Indeterminate forms
• L'Hôpital's rule states that the limit of a fraction in indeterminate form equals the limit of the ratio of its derivatives
• What is the other common indeterminate form to which L'Hôpital's rule can be applied?

  $\frac{\infty}{\infty}$
• L'Hôpital's rule cannot be used if the derivative of the denominator is zero
• Match the indeterminate form with its description:

  $\frac{0}{0}$ ↔️ Quotient of two functions that both approach 0

  $\frac{\infty}{\infty}$ ↔️ Quotient of two functions that both approach infinity

  $0 \cdot \infty$ ↔️ Product of a function approaching 0 and infinity

  $1^{\infty}$ ↔️ Exponential function with base 1 and infinite exponent
• What is the result of applying L'Hôpital's rule to $\lim_{x \to 0} \frac{\sin(x)}{x}$?


  1
• Conditions for applying L'Hôpital's rule

  1️⃣ $\lim_{x \to c} f(x)$ and $\lim_{x \to c} g(x)$ both approach 0 or $\pm \infty$

  2️⃣ $f'(x)$ and $g'(x)$ exist

  3️⃣ $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ exists
• What is the indeterminate form $\frac{\infty}{\infty}$?


  Quotient of two infinities
• Applying L'Hôpital's Rule to $\lim_{x \to 0} \frac{\sin(x)}{x}$ results in 1.


  True
• Steps to apply L'Hôpital's Rule

  1️⃣ Verify the indeterminate form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

  2️⃣ Differentiate the numerator and denominator separately.

  3️⃣ Calculate the new limit of the derivative ratio.

  4️⃣ If the new limit is indeterminate, repeat differentiation.
• What is the value of $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$ using L'Hôpital's Rule?


  2
• What is the first step in applying L'Hôpital's Rule to $\frac{0}{0}$?


  Check the indeterminate form
• What is the value of $\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$ using L'Hôpital's Rule?


  4
• The indeterminate form $\frac{0}{0}$ arises when the quotient of two functions both approach zero
• What are indeterminate forms in calculus?

  Limits needing special evaluation
• What is the key idea behind L'Hôpital's Rule?

  Ratio of derivatives
• Conditions for applying L'Hôpital's Rule

  1️⃣ Both $f(x)$ and g(x)</latex> approach 0 or $\pm \infty$

  2️⃣ $f'(x)$ and $g'(x)$ exist

  3️⃣ $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ exists
• L'Hôpital's Rule involves taking the derivatives of the numerator and denominator


  True
• When encountering the indeterminate form $\frac{0}{0}$, L'Hôpital's Rule requires differentiating both the numerator and the denominator
• L'Hôpital's Rule can be applied to $\frac{\infty}{\infty}$ indeterminate forms


  True
• When you encounter a limit that results in the \frac{\infty}{\infty} indeterminate form, L'Hôpital's Rule can be applied.
• What is the first step when evaluating \lim_{x \to \infty} \frac{x^{2} + 2x}{x^{2} +$3x +$ 1} using L'Hôpital's Rule?


  Verify indeterminate form
• The \frac{0}{0}</latex> indeterminate form arises when the numerator and denominator both approach 0.
• Indeterminate forms are limits that cannot be evaluated directly using algebraic methods
• To evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using L'Hôpital's rule, we take the derivative of both the numerator and the denominator
• What are the most common indeterminate forms that L'Hôpital's rule can address?

  $\frac{0}{0}$ and $\frac{\infty}{\infty}$
• Steps for applying L'Hôpital's rule:

  1️⃣ Verify that the limit is an indeterminate form

  2️⃣ Take the derivative of the numerator

  3️⃣ Take the derivative of the denominator

  4️⃣ Evaluate the new limit
• What type of values do $f(x)$ and $g(x)$ approach as $x$ approaches $c$?


  0 or $\pm \infty$
• The limit $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ must exist for L'Hôpital's Rule to be valid
• The limit $\lim_{x \to 0} \frac{\sin(x)}{x}$ is equal to 1.


  True
• Steps to apply L'Hôpital's Rule to $\frac{0}{0}$ indeterminate forms:


  1️⃣ Check for $\frac{0}{0}$ form

  2️⃣ Find the derivatives $f'(x)$ and $g'(x)$

  3️⃣ Evaluate $\lim_{x \to c} \frac{f'(x)}{g'(x)}$

  4️⃣ Repeat if needed
• What is a common mistake when using L'Hôpital's Rule?

  Forgetting to check the indeterminate form
• L'Hôpital's Rule can be used to evaluate limits that are in the $\frac{0}{0}$ form.


  True
• When encountering the $\frac{\infty}{\infty}$ indeterminate form, L'Hôpital's Rule states that you can differentiate the numerator and denominator
• Steps to apply L'Hôpital's Rule to $\frac{\infty}{\infty}$ indeterminate forms:


  1️⃣ Verify the indeterminate form

  2️⃣ Differentiate $f'(x)$ and $g'(x)$

  3️⃣ Apply L'Hôpital's Rule

  4️⃣ Repeat if needed
• The limit \lim_{x \to \infty} \frac{x^{2} + 2x}{x^{2} +$3x +$ 1} is equal to 1
• L'Hôpital's Rule is applied to evaluate the limit $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ if it is still in the \infty
• Why cannot L'Hôpital's Rule be applied to \lim_{x \to 0} \frac{x^{2} - 1}{x - 1}</latex>?

  The denominator does not approach 0
• The limit \lim_{x \to 0} \frac{e^{x} - 1}{x}</latex> equals 1 using L'Hôpital's Rule.


  True
• L'Hôpital's rule can be applied to the indeterminate form $\frac{0}{0}$


  True