4.7 Using L'Hôpital's Rule

Cards (93)

  • L'Hôpital's Rule can only be applied to the indeterminate forms \frac{0}{0}</latex> and ∞∞\frac{\infty}{\infty}.

    False
  • What does L'Hôpital's Rule state in its simplest form?
    Limit of fraction equals derivative of numerator over derivative of denominator
  • What are indeterminate forms in calculus?
    Expressions whose values cannot be determined directly
  • What are the three conditions for applying L'Hôpital's Rule?
    Indeterminate form, continuity, differentiability
  • What is an example of the indeterminate form 0/0?
    lim⁡x→0xsin⁡(x)\lim_{x \to 0} \frac{x}{\sin(x)}
  • What is the value of lim⁡x→0xsin⁡(x)\lim_{x \to 0} \frac{x}{\sin(x)}?

    1
  • After applying L'Hôpital's Rule to x2ex\frac{x^{2}}{e^{x}} once, the new expression is \frac{2x}{e^{x}}
  • What is the final step in evaluating the limit after applying L'Hôpital's Rule?
    Check for indeterminate forms again
  • What is L'Hôpital's Rule used for?
    Evaluating limits of indeterminate forms
  • L'Hôpital's Rule can be applied repeatedly if the resulting limit is still an indeterminate form
  • To use L'Hôpital's Rule, you first take the derivative of the numerator and the derivative of the denominator.
  • Indeterminate forms always require L'Hôpital's Rule to determine their limit values.
    False
  • List the seven common indeterminate forms in calculus.
    00,∞∞,0×∞,∞−∞,00,1∞,∞0\frac{0}{0}, \frac{\infty}{\infty}, 0 \times \infty, \infty - \infty, 0^{0}, 1^\infty, \infty^{0}
  • To which indeterminate forms can L'Hôpital's Rule be directly applied?
    00\frac{0}{0} and \frac{\infty}{\infty}</latex>
  • L'Hôpital's Rule can be used to determine the actual limit values of indeterminate forms.
    True
  • What is the value of \lim_{x \to \infty} \frac{x^{2}}{e^{x}}</latex>?
    0
  • Steps for applying L'Hôpital's Rule to lim⁡x→0x2sin⁡(x)\lim_{x \to 0} \frac{x^{2}}{\sin(x)}

    1️⃣ Identify the numerator and denominator.</step_end><step_start>Find the derivative of the numerator, f′(x)f'(x).
    2️⃣ Find the derivative of the denominator, g′(x)g'(x).
    3️⃣ Evaluate the limit of f′(x)g′(x)\frac{f'(x)}{g'(x)} as x→0x \to 0.
  • To evaluate the limit of a differentiated expression, you first write the differentiated ratio
  • Steps to evaluate the limit of a differentiated expression
    1️⃣ Write the differentiated expression
    2️⃣ Substitute the value to which x approaches
    3️⃣ Check for indeterminate forms
    4️⃣ Simplify if necessary
  • L'Hôpital's Rule states that the limit of a fraction resulting in an indeterminate form is equal to the limit of the derivatives
  • The indeterminate form \(0 \times \infty\) arises when one factor approaches 0 while the other approaches infinity
  • What is the derivative of \(x\)?
    1
  • What is the limit limx→0xsin⁡(x)lim_{x \to 0} \frac{x}{\sin(x)}?

    1
  • What is the limit limx→∞x2exlim_{x \to \infty} \frac{x^{2}}{e^{x}}?

    0
  • What is the first step in applying L'Hôpital's Rule to evaluate a limit?
    Identify numerator and denominator
  • What is the limit of \lim_{x \to 0} \frac{x^{2}}{\sin(x)}</latex> after applying L'Hôpital's Rule?
    0
  • What is the result of substituting x=x =0 0 into 2xcos⁡(x)\frac{2x}{\cos(x)}?

    0
  • The indeterminate form 0 × ∞ arises when one factor approaches 0 and the other approaches infinity.

    True
  • What type of indeterminate form results from infinity minus infinity?
    ∞ - ∞
  • Match the indeterminate form with its description:
    ∞ - ∞ ↔️ Both terms approach infinity
    0^0 ↔️ Base and exponent approach 0
    1^∞ ↔️ Base approaches 1, exponent approaches infinity
    ∞^0 ↔️ Base approaches infinity, exponent approaches 0
  • One condition for applying L'Hôpital's Rule is that the limit must be in the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

    True
  • When applying L'Hôpital's Rule to \(\lim_{x \to 0} \frac{\sin(x)}{x}\), the resulting limit is 1
  • Match the indeterminate form with its example:
    \(\frac{0}{0}\) ↔️ \(\lim_{x \to 0} \frac{x}{\sin(x)}\)
    \(\frac{\infty}{\infty}\) ↔️ \(\lim_{x \to \infty} \frac{x^2}{e^x}\)
    0 × ∞ ↔️ \(\lim_{x \to 0} x \ln(x)\)
    \(\infty - \infty\) ↔️ \(\lim_{x \to \infty} (x - x^2)\)
  • L'Hôpital's Rule is the only method for evaluating indeterminate forms.
    False
  • When evaluating the limit of a differentiated expression after applying L'Hôpital's Rule, the first step is to substitute the value to which \( x \) approaches
  • What is the first step in evaluating the limit after applying L'Hôpital's Rule?
    Substitute the value
  • Steps to evaluate \lim_{x \to 0} \frac{2x}{\cos(x)}</latex>
    1️⃣ Substitute x=x =0 0
    2️⃣ Evaluate the expression
    3️⃣ Conclude the limit value
  • Applying L'Hôpital's Rule twice to x2ex\frac{x^{2}}{e^{x}} results in a limit of 00.

    True
  • Match the indeterminate form with its transformation method:
    0×∞0 \times \infty ↔️ 01∞\frac{0}{\frac{1}{\infty}}
    ∞−∞\infty - \infty ↔️ Combine terms and factor
    000^{0} ↔️ ln⁡y=\ln y =g(x)ln⁡f(x) g(x) \ln f(x)
  • The limit of (1+1x)x\left(1 + \frac{1}{x}\right)^{x} as x→∞x \to \infty is e