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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.

ຕົວຢ່າງ​ທີ່​ໄດ້​ເຮັດ · 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}

ຂັ້ນ​ຕອນ​ຕໍ່​ຂັ້ນ​ຕອນ

  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.

ເປີດ​ເຜີຍ​ຄຳ​ຕອບ
1

ປັດຈຸບັນ​ທ່ານ ເລືອກ​ບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມ​ເອງ. ທຸກ​ບາດກ້າວ, ຮູບ, ຄໍາຕອບ​ຖືກ​ລົບ​ຈົນ​ກວ່າ​ເຈົ້າ​ຈະ​ຖາມ​ມັນ.

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ວິທີການ: 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.

ຄໍາຖາມທີ່ຄົນຖາມ

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.

ເພີ່ມເຕີມໃນ 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