Learning Outcomes

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

Textbook

  • Chapter 4.8   L’HĂ´pital’s Rule

Textbook Assignment

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

WeBWork Assignment

  • Application-LHopitals Rule

Exit problems of the session 

  1. Evaluate the limits for the following function:

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

 

Key Concepts

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

\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:  rewrite the function to form indeterminate quotient  \dfrac{0}{0}  or  \dfrac{\infty}{\infty},  then apply L’HĂ´pital’s rule.
  • Indeterminate power 0^0, \infty^0, 1^\infty:   apply ln to the function and rewrite the function to form indeterminate quotient  \dfrac{0}{0}  or  \dfrac{\infty}{\infty},  then apply L’HĂ´pital’s rule. Raise the result to the power of e to get the limit of the indeterminate power. 

 

Videos and Practice Problems of Selected Topics

  1. \rhd L’HĂ´pital’s Rule (8:52)  Indeterminate forms and an introduction to L’HĂ´pital’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: L’Hôpital’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: L’Hôpital’s Rule: \infty/\infty
  6. \rhd Composite exponential functions (13:10)  Find the limit of indeterminate power \displaystyle\lim_{x\to 0^+}(\sin x)^{\frac{1}{\ln x}}