Fundamental Theorems on Limits

Theorem III. The limit of the algebraic sum of finitely many functions equals the sum of the limits of these functions:

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Theorem IV. The limit of the product of two functions equals the product of the limits of these functions:

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Theorem V. The limit of the ratio of two functions equals the quotient of the limits of the numerator and the denominator:

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Theorem VI (the sandwich theorem). If, in a neighborhood of a point а, the inequalities u (x) £ z (x) £ v (x) hold , then limit of z(x) exists and equals .

Theorem VII. If a function f(x) is monotonically increases (decreases) and is bounded as х® а above (below), then f(x) has a finite limit .

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