Fundamental Theorems on Limits

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

 

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

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

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Computations of limits. Examples.

I. Limits as x ;.

(1)

The limits in the numerator and the denominator equal zero.

To find the limit of a linear-fractional function, we must divide the numerator and the denominator by х to the maximum power among the powers of x in the numerator and the denominator.

(2)

because х ⁴ is the maximum power of x in the numerator and the denominator.

(3) (divide by х ²).

A simple method for finding limits of linear-fractional functions as х  is to leave the term containing the maximum power of х in the numerator and the denominator:

4) ,

 

5) ,

 

6) .

 

Let us find limits (1), (2), (3) by the simple method:

,

 

,

 

.

 

Deleting the terms containing lower powers of x from the numerator and the denominator is only possible because, after division by х to the maximum power, the limits of all such terms vanish.

II. Limits as х  а. Looking for a limit, first, substitute in the function. If we obtain a number, then this number is the limit of the function. If we obtain one of the indeterminacies ,1^, and , then we must eliminate it by transforming the function and then to pass to the limit.

 

(1) ,

 

 

(2) ,

 

(3) ,