List of mathematical formulas: “L’Hôpital’s rules

The main formulas and rules associated with L’Hôpital’s rule are:

  1. Statement of L’Hôpital’s rule:
    If the limit (\lim_{x \to c} \frac{f(x)}{g(x)}) takes the indefinite form (\frac{0}{0}) or (\frac{\infty}{\infty}), then:
    [
    \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
    ]
    provided that the limit on the right exists or is equal to (\pm \infty).
  2. Application of the rule:
  • Check that both functions (f(x)) and (g(x)) are differentiable in a neighborhood of point (c) (except, possibly, point (c) itself).
  • If after the first application of L’Hôpital’s rule you still get an indefinite form, you can apply the rule again.
  1. Forms in which the rule applies:
  • (\frac{0}{0})
  • (\frac{\infty}{\infty})
  1. Additional cases:
    If the limit is (\lim_{x \to c} f(x) = \infty) and (\lim_{x \to c} g(x) = 0), or vice versa, one can use:
    [
    \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \quad \text{(if applicable)}
    ]
  2. Example:
    For the function (\lim_{x \to 0} \frac{\sin x}{x}):
  • We apply L’Hôpital’s rule:
    [
    \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1
    ]

От Math

Добавить комментарий

Ваш адрес email не будет опубликован. Обязательные поля помечены *