A limit asks what a function approaches, not what it equals. Try substitution first; if it gives 0/0, factor and cancel, or differentiate top and bottom (L'Hôpital). The graph shows the function near the point with the limit marked.
Exemple résolu · limit of sin(x)/x as x -> 0
Étape par étape
- \lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}
Try direct substitution first.
- \frac{0}{0}
Substitution gives 0/0: an indeterminate form.
- \lim \frac{\cos{\left(x \right)}}{1}
L'Hôpital's rule: differentiate numerator and denominator separately.
- = 1
Substitute.
Révèle la réponse
À vous Choisissez un problème, ou tapez ou dessinez le vôtre. Chaque étape, une image, la réponse cachée jusqu'à ce que vous le demandiez.
Un compte gratuit ajoute des notes sur chaque leçon, un enregistrement de ce que vous avez terminé, vos problèmes résolus en un seul endroit, et un tuteur que vous pouvez demander sur cette page. Les maths eux-mêmes sont ouverts à tout le monde, signé ou non.
Inscription ConnexionSymboles utilisés ici
Appuyez sur n'importe quel symbole pour la définition complète, une image et ce que signifie chaque lettre.
Comment faire: Limits
- Try direct substitution first.
- Substitution gives 0/0: an indeterminate form.
- L'Hôpital's rule: differentiate numerator and denominator separately.
- Substitute.
Questions posées par les gens
What is a derivative in one sentence?
The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.
What is an integral in one sentence?
The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.
Why are derivatives and integrals opposites?
That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.
When do I use substitution and when integration by parts?
Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.
Plus en Calculus
DerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests