Limits of Functions

1. Properties of Limits of Functions
   For two functions and , where the limits and exist:
1. (where is a constant)
5. (where )


2. Calculating the Limit of a Function
1. Indeterminate form :
1. If both the numerator and denominator are polynomials, factorize both and then cancel common factors.
2. If either the numerator or denominator involves a radical, rationalize the radical expression.
2. Indeterminate form :
1. If (degree of numerator) (degree of denominator), the limit is the ratio of the leading coefficients.
2. If (degree of numerator) (degree of denominator), the limit is .
3. If (degree of numerator) (degree of denominator), the function diverges to or .


3. Indeterminate form :
1. For polynomials, factor out the highest degree term.
2. For expressions involving radicals, rationalize the side with the radical.
4. Indeterminate form :
   Convert into forms such as , , , or (where is a constant).


3. Determining Unknown Coefficients
   For two functions and :
1. If (where is a real number) and , then .
2. If (where is a non-zero real number) and , then .


4. Comparison of Limits of Functions
   For three functions , , , and for all real numbers close to :
1. If and both and exist, then .
2. If and (where is a real number), then