## 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} = L^{n}

### 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.