Prove that the limit of √(n+√n) - √n = 1/2 (ILIEKMATHPHYSICS)

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This video references the book "Introduction to Real Analysis" by Bartle and Sherbert (Fourth Edition). For more details regarding the techniques used in this video, see Section 2.4 and Section 3.1.

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Комментарии
Автор

Factor sqrt(n) from the expression, to get sqrt(n) ( sqrt(1 + 1/sqrt(n) ) - 1).

Let h = 1/sqrt(n).

lim n -> infinity sqrt(n) (sqrt(1-1/sqrt(n)) -1) = lim h -> 0+ (sqrt(1 + h) -1) / h.

This is, up to the one-sided limit, the definition of the derivative of sqrt(x) evaluated at x = 1.

The two-sided limit exists and it is 1/2.

This method pushes the formal limit argument onto the proving the derivative.

PrincetonAthleticClub
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fantastic video! keep up the great work :)

DemigodTet
Автор

Change to a fraction with (1/n) denominator, then apply L'H -simples!

johnpaterson