l'Hospital's rule

l'Hospital's rule

Remember that one tricky Limits section that required intense algebraic manipulation to avoid getting 0/0 or infinity/infinity limits? We will now revisit it again, but with the knowledge of derivatives. In this section, we will learn how derivatives enable us to efficiently evaluate the limits of a function using the "L'Hospital's rule".
Basic Concepts: Finding limits algebraically - direct substitution , Limit laws

Lessons

Note *l'Hôpital's Rule applies to 2 types of indeterminate forms:

type 00\frac{0}{0} (that is, lim⁡\limx →c c f(x)=0f(x)=0 and lim⁡\limx →c cg(x)=0g(x)=0)
or
type ∞∞\frac{\infty}{\infty} (that is, lim⁡\limx →c c f(x)=±∞f(x)=\pm \infty and lim⁡\limx →c cg(x)=±∞g(x)=\pm \infty)

Then according to l'Hôpital's Rule: lim⁡\limx →c c f(x)g(x)=\frac{f(x)}{g(x)}= lim⁡\limx →c c f′(x)g′(x)\frac{f'(x)}{g'(x)}
  • Introduction
    Evaluating the limit of the form:
    lim⁡\limx →c c f(x)g(x)\frac{f(x)}{g(x)}
  • 1.
    Evaluating the limit.
    Find:
    a)
    lim⁡\limx →1 1 ln⁡xx−1\frac{\ln x}{x-1}

    b)
    lim⁡\limx →∞ \infty ln⁡xx−1\frac{\ln x}{x-1}

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3.
Applications of the Derivative
3.1
Position velocity acceleration
3.2
Critical number & maximum and minimum values
3.3
l'Hospital's rule
3.4
Curve sketching
3.5
Related rates
3.6
Mean value theorem
3.7
Linear approximation
CLEP Calculus
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