Rules for Limits

Definition of Limit: If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then the limit of f(x) , as x approaches c, is L.

This is written as . Note that x is not set equal to c; in fact the function need not be even defined at c.

Limit from the left: If f(x) becomes arbitrarily close to a single number L as x approaches c from the left, then we write .

Limit from the right: If f(x) becomes arbitrarily close to a single number L as x approaches c from the right, then we write .

Theorem on limits: If f is a function and c and L are real numbers, then

if and only if the limit from the left and the limit from the right are each equal to L, or

and .

Limit of a constant, b: 

Limit of x: 

Limit of [constant * f(x)]: 

Limit of a sum: , or the limit of a sum is the sum of the limits.

Limit of a product: , or the limit of a product is the product of the limits.

Limit of a quotient: , provided that , or the limit of a quotient is the quotient of the limits.

Limit of a power: 

Limit at Infinity for Rational Functions: Consider the functions f (x) and g(x), both of which are polynomials. The largest power of x in f is n. The largest power of x in g is m.

Let  be the function which is the ratio of f to g.

As , the limit of R is .

L can be determined by considering the largest powers, m and n.

If the largest power of x is in the denominator, or m > n, the denominator increases faster than the numerator, and L is 0.

If the largest power of x is in the numerator, or n > m, the numerator increases faster and
L approaches .

If the m = n, L equals the ratio of the coefficients of those largest powers:

For example,