Learning Outcomes
- Recognize when to apply L’Hôpital’s rule.
- Identify indeterminate forms produced by quotients, products, and powers, and apply L’Hôpital’s rule in each case.
Textbook
- Chapter 4.8 L’Hôpital’s Rule
Textbook Assignment
- p. 470: 356, 359, 362-367 all, 373, 377, 380, 387, 393, 395
WeBWork Assignment
- Application-LHopitals Rule
Exit problems of the session
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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’Hôpital’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’Hôpital’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’Hôpital’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
- $\rhd$ L’Hôpital’s Rule (8:52) Indeterminate forms and an introduction to L’Hôpital’s Rule.
- $\rhd$ Limit at 0 (7:42) Find $\displaystyle\lim_{x\to 0} \dfrac{2\sin(x) – \sin (2x)}{x-\sin(x)}$.
- * Practice: L’Hôpital’s Rule: $0/0$
- $\rhd$ Limit at infinity (5:15) Find $\displaystyle\lim_{x\to \infty} \dfrac{4x^2-5x}{1-3x^2}$.
- * Practice: L’Hôpital’s Rule: $\infty/\infty$
- $\rhd$ Composite exponential functions (13:10) Find the limit of indeterminate power $\displaystyle\lim_{x\to 0^+}(\sin x)^{\frac{1}{\ln x}}$
