Math Help

In a Nut Shell: The notion of the limit of a function occurs when you are interested in the

value of the function, say y(x), when x takes on a certain value. It is an important concept

in differential calculus. The idea of the limit of a function, y(x), is as follows:

(The statement below is the Informal Definition of a Limit)

The number L is the limit of y(x) as x approaches a provided one can

make the number y(x) as close to L as one pleases by merely picking x

sufficiently near, but not equal to, the number a.

Consider the graph, y(x) shown below. Let y(x) be a “smooth” curve with no “missing”

points on the curve. Here the term “smooth” is a very loose description of the nature of

the curve. It relates to “continuity” of a function which will be described in a later section.

In this case, y(x) takes on the value, L, as x approaches a. This limit is

normally expressed as follows:

lim y(x) = L

x → a

which reads as the limit of y of x as x approaches a takes on the value of L.

Squeeze Law Suppose that f(x) ≤ g(x) ≤ h(x) for all x ≠ a in some neighborhood

of a and also that

lim f(x) = L = lim h(x)

x → a x → a

then lim g(x) = L

x → a

Suppose g(x) is a complicated function and you can find two simpler functions, f(x) and

h(x) that bound g(x). Then you can use limits on f(x) and h(x) to find the limit on g(x).