In these rules let "a", "A", and "B" be real numbers and "f" and "g" be functions such that limit as x approaches a of f of x equals A and limit as x approaches a of g of x equals B

 

Limit of a constant:

 

limit as x approaches a of the constant k equals k

 

Limit of a constant times a function:

 

limit as x approaches a of the constant k times the function f of x equals k times the limit as x approaches a of f of x which equals k times A

 

Limit of the sum or difference of functions:

 

limit as x approaches a of open bracket f of x plus or minus g of x closed bracket equals limit as x approaches a of f of x plus or minus limit x approaches a of g of x equals A plus or minus B

 

Limit of the product of functions:

 

limit as x approaches a of open bracket f of x times g of x closed bracket equals limit as x approaches a of f of x times limit as x approaches a of g of x which equals A times B

 

Limit of the quotient of functions (B cannot be equal to zero):

 

limit as x approaches a of the fraction f of x over g of x equals the limit as x approaches a of f of x divided by limit as x approaches a of g of x which equals A divided by B

 

Limit of a polynomial function:

 

limit as x approaches a of the polynomial function p of x equals p of a

 

Limit of a function raised to an exponent (k is any real number):

 

limit as x approaches a of open bracket f of x closed bracket raised to the constant k equals open bracket limit as x approaches a of f of x closed bracket raised to k which equals A raised to the k

 

Equality of the limits of two functions:

 

limit as x approaches a of f of x equals limit as x approaches a of g of x if f(x) = g(x) for all x cannot be equal to a

 

Limit of an exponential expression with a function as the exponent (b is any real number > 0):

 

limit as x approaches a of the constant b raised to the function f of x equals b raised to the limit as x approaches a of f of x which equals b raised to A

 

Limit of a logarithm of a function (b is any real number > 0 except for 1):

 

limit as x approaches a of open bracket log base b of f of x closed bracket equals log base b of open bracket limit as x approaches a of f of x closed bracket which equals log base b of A

 

Limits at Infinity (n is any positive real number):

 

limit as x approaches infinity of open bracket 1 over x to the n power closed bracket equals zero  and  limit as x approaches negative infinity of open bracket 1 over x to the n power closed bracket equals zero

 

Using limits to find the instantaneous rate of change (also known as the derivative):

 

f prime of x equals limit as h approaches zero of the fraction f of (x plus h) minus f of x all over h