Limit Facts

For these facts, assume lim_x−>c f(x) = L and lim_x−>c g(x) = K

Scalar Product

lim_x−>c b f(x) = bL

Sum and Difference

lim_x−>c f(x)+g(x) = L+K
lim_x−>c f(x)-g(x) = L-K

Product and Quotient

lim_x−>c f(x) g(x) = L K

Power

lim_x−>c f(x)ⁿ = Lⁿ

Polynomial Functions

By combining the scalar product, sum, and power rules, we get this:

If p is a polynomial function and c is a real number then
lim_x−>c p(x) = p(c)

Rational Functions

A rational function r(x) is the ratio of two polynomial functions.

If r is a rational function r(x) = p(x)/q(x), and c is real, and q(c) ≠ 0, then
lim_x−>c r(x) = r(c) = p(c)/q(c).

If q(c)=0 and p(c) ≠ 0 then the rational function is unbounded, and the limit does not exist.

If q(c)=0 and p(c)=0 as well, then divide both p(x) and q(x) by the monomial (x-c).  

The new rational function r'(x) = (p(x)/(x-c)) / (q(x)/(x-c)) is equal to r(x) for all values of x except x=c, so the Indeterminate Form rules can be used.

Composition of Functions

If lim_x−>c g(x) = K and lim_x−>K f(x) = L then lim_x−>c f(g(x)) = L 

Missing Point

If f(x)=g(x) for all x≠c in an open interval containing c, and lim_x−>c f(x)=L then lim_x−>c g(x)=L

Indeterminate Form

If lim_x−>c f(x)=0, and lim_x−>c g(x)=0, then lim_x−>c f(x)/g(x) is said to have "Indeterminate Form"

Similarly, if lim_x−>c f(x)=∞, and lim_x−>c g(x)=∞, then lim_x−>c f(x)/g(x) is also said to have "Indeterminate Form"

If a new function h(x) can be found such that h(x)=f(x)/g(x) for all x≠c in an open interval containing c, then the "Missing Point" rule can be used to solve the Indeterminate Form.  Possible methods of finding such a function are:

1. Factor the numerator and denominator, and cancel common factors -- in particular, cancel the factor (x-c)

2. "Rationalize" the numerator, if it contains a radical.  For example,

(sqrt(x+1)-1)/x

can be rationalized as

1/(sqrt(x+1)+1)

3. Using L'Hopital's Rule.

Squeeze Theorem

If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c (except possibly at c itself) and
if lim_x−>c h(x) = L = lim_x−>c g(x),
then lim_x−>c f(x) = L

Related pages in this website

Introduction to limits

Graphical explanation of limits

The webmaster and author of this Math Help site is Graeme McRae.