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The general form of L'Hôpital's rule covers many cases. Let c and L be extended real numbers: real numbers, positive or negative infinity. Let I be an open interval containing c (for a two-sided limit) or an open interval with endpoint c (for a one-sided limit, or a limit at infinity if c is infinite).
L'Hôpital's Rule. L'Hôpital's Rule can help us calculate a limit that may otherwise be hard or impossible. L'Hôpital is pronounced "lopital". He was a French mathematician from the 1600s.
Guillaume François Antoine, Marquis de l'Hôpital[1] (French: [ɡijom fʁɑ̃swa ɑ̃twan maʁki də lopital]; sometimes spelled L'Hospital; 1661 – 2 February 1704) [a] was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞.
This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.
L'Hôpital's rule is a theorem used to find the limit of certain types of indeterminate forms; indeterminate forms are expressions that result from attempting to compute a limit through use of substitution.
L’Hospital’s Rule works great on the two indeterminate forms 0/0 and ±∞/±∞ ± ∞ / ± ∞. However, there are many more indeterminate forms out there as we saw earlier. Let’s take a look at some of those and see how we deal with those kinds of indeterminate forms. We’ll start with the indeterminate form (0)(±∞) (0) (± ∞).
L'Hopital's Rule. Suppose \(f\) and \(g\) are differentiable functions such that \(g'(x) \neq 0\) on an open interval \(I\) containing \(a;\)
L’Hopital’s Rule. Consider the limit lim x → a f(x) g(x). If both the numerator and the denominator are finite at a and g(a) ≠ 0, then lim x → a f(x) g(x) = f(a) g(a). Example. lim x → 3 x2 + 1 x + 2 = 10 5 = 2.
L’Hospital’s rule is a way to figure out some limits that you can’t just calculate on their own. Precisely, to estimate the limit of a fraction that, when it gives 0/0 or ∞/∞, we often use L’Hopital’s rule.
Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits. About Khan Academy: Khan Academy offers practice exercises, instructional videos...