maths.free › Calculus › Limits

Limits

Substitution, factoring and L'Hôpital's rule.

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.

Pracovný príklad · limit of sin(x)/x as x -> 0

Calculus · Limits

Limit of sin(x)/x as x → 0

\lim_{x \to 0} \frac{\sin{\left(x \right)}}{x}

Krok za krokom

  1. \lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}

    Try direct substitution first.

  2. \frac{0}{0}

    Substitution gives 0/0: an indeterminate form.

  3. \lim \frac{\cos{\left(x \right)}}{1}

    L'Hôpital's rule: differentiate numerator and denominator separately.

  4. = 1

    Substitute.

Odhaliť odpoveď
1

Teraz ty Vyberte si problém, alebo napíšte alebo nakreslite vlastný.Každý krok, obrázok, odpoveď skrytá, kým sa neopýtate.

Udržujte svoje vlastné pracovné

Bezplatný účet pridáva poznámky ku každej lekcii, záznam o tom, čo ste dokončili, vaše vyriešené problémy na jednom mieste, a učiteľa, ktorého sa môžete opýtať na túto stránku.Math samotná je otvorená pre každého, prihlásený alebo nie.

Zaregistrovať sa Prihlásenie

Symboly použité tu

Ťuknutím na ľubovoľný symbol zobrazíte úplnú definíciu, obrázok a význam každého písmena.

Ako nakupovať: Limits

  1. Try direct substitution first.
  2. Substitution gives 0/0: an indeterminate form.
  3. L'Hôpital's rule: differentiate numerator and denominator separately.
  4. Substitute.

Otázky, ktoré ľudia kladú

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.

Viac v kategórii 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