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. 


  • 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 $\displaystyle\lim_{x\to 0^+}(\sin x)^{\frac{1}{\ln x}}$