Lecture4 thelimitofafunction 140915141659 phpapp02, Lecture notes of Applied Mathematics

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2.2 The Limit of a Function

The Limit of a Function

Let’s investigate the behavior of the function f defined by

f ( x ) = x^2 – x + 2 for values of x near 2.

Limit of a Function - Graphically

From the table and the graph of f (a parabola) shown in Figure 1 we see that when x is close to 2 (on either side of 2), f ( x ) is close to 4.

Figure 1

The Limit of a Function

As x gets close to 2, f ( x ) gets close to 4

“the limit of the function f ( x ) = x^2 – x + 2 as x approaches 2 is equal to 4.”

The notation for this is

The Limit of a Function

An alternative notation for

is f ( x )  L as x  a

which is usually read “ f ( x ) approaches L as x approaches a. ”

Notice the phrase “but x  a ” in the definition of limit. This means that in finding the limit of f ( x ) as x approaches a, it does not matter what is happening at x = a.

In fact, f ( x ) need not even be defined when x = a. The only thing that matters is how f is defined near a.

The Limit of a Function

Figure 2 shows the graphs of three functions. Note that in part (c), f ( a ) is not defined and in part (b), f ( a )  L.

But in each case, regardless of what happens at a , it is true that lim x  a f ( x ) = L.

Figure 2 in all three cases

Example 1 - Graphically

Example 1 – Guessing a Limit from Numerical Values

Guess the value of

Solution:

F(1) is undefined, but that doesn’t matter because the

definition of lim x  a f ( x ) says that we consider values of x that

are close to a but not equal to a.

Finding Limits - Examples

One-Sided Limits

One-Sided Limits

“ t  0 – ” values of t that are less than 0  “left

“ t  0 +” values of t that are greater than 0  “right

One-Sided Limits

Notice that Definition 2 differs from Definition 1 only in that

we require x to be less than a.

Similar definition for right-handed limit

Example 5 – One-Sided Limits from a Graph

The graph of a function g is shown in Figure 10. Use it to state the values (if they exist) of the following:

Figure 10

Example 5 – Solution

From the graph we see that the values of g ( x ) approach 3 as x approaches 2 from the left, but they approach 1 as x approaches 2 from the right.

Therefore

and

(c) Since the left and right limits are different, we conclude from (3) that lim x  2 g ( x ) does not exist.