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Learning Outcomes

1. Recognize when to apply L’HĂ´pital’s rule.
2. Identify indeterminate forms produced by quotients, products, and powers, and apply Lâ€™HĂ´pitalâ€™s rule in each case.Â

Textbook

• Chapter 4.8 Â  Lâ€™HĂ´pitalâ€™s Rule

Textbook Assignment

• p. 470: Â  Â 356, 362, 367, 370, 371, 377, 387, (393, 395 optional)

WeBWork Assignment

• Application-LHopitals Rule

Exit problemsÂ of the sessionÂ

1. Evaluate the limits for the following function:

(a). Â $\displaystyle\lim_{x\to 0}\dfrac{\sin x-\tan x}{x}$ Â  Â  Â  Â (b). Â  $\displaystyle\lim_{x\to 1}\dfrac{ x-1}{\ln x}$ Â  Â  Â  Â (c). Â  $\displaystyle\lim_{x\to \infty}x^3e^{-2x}$ Â  Â  Â  Â (d). Â  $\displaystyle\lim_{x\to \infty}x^{1/x}$

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Key Concepts

• Lâ€™HĂ´pitalâ€™s rule’s can be used to evaluate the limit of a quotient when the indeterminate form Â $\dfrac{0}{0}$ Â or Â $\dfrac{\infty}{\infty}$ Â arises. Â In these two cases:Â

$\displaystyle\lim_{x\to a}\dfrac{f(x)}{g(x)}=\displaystyle\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$

• Indeterminate product $0\cdot \infty$: Â rewrite the function to form indeterminate quotient Â $\dfrac{0}{0}$ Â or Â $\dfrac{\infty}{\infty}$, Â then apply Lâ€™HĂ´pitalâ€™s rule.
• Indeterminate power $0^0, \infty^0, 1^\infty$: Â  apply $ln$ to the function and rewrite the function to formÂ indeterminate quotient Â $\dfrac{0}{0}$ Â or Â $\dfrac{\infty}{\infty}$, Â then apply Lâ€™HĂ´pitalâ€™s rule. Raise the result to the power of $e$ to get the limit of the indeterminate power.Â

#### Videos and Practice Problems of Selected Topics

1. $\rhd$ L’HĂ´pitalâ€™s Rule (8:52) Â Indeterminate forms and an introduction toÂ Lâ€™HĂ´pitalâ€™s Rule.
2. $\rhd$ Limit at 0 (7:42) Find $\displaystyle\lim_{x\to 0} \dfrac{2\sin(x) – \sin (2x)}{x-\sin(x)}$.
3. * Practice: Lâ€™HĂ´pitalâ€™s Rule: $0/0$
4. $\rhd$ Limit at infinity (5:15) Find $\displaystyle\lim_{x\to \infty} \dfrac{4x^2-5x}{1-3x^2}$.
5. * Practice: Lâ€™HĂ´pitalâ€™s Rule: $\infty/\infty$
6. $\rhd$ Composite exponential functions (13:10) Â Find the limit of indeterminate power $\displaystyle\lim_{x\to 0^+}(\sin x)^{\frac{1}{\ln x}}$