## The Limit of a Function

To find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them.

*Let’s investigate the behavior of the function f defined by*
*f(x) = x*^{2} *– x + 2 for values of x near 2.*

## The Limit of a Function

*The following table gives values of f(x) for values of x close *
to 2 but not equal to 2.

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

## The Limit of a Function

## The Limit of a Function

*In fact, it appears that we can make the values of f(x) as *
*close as we like to 4 by taking x sufficiently close to 2.*

We express this by saying “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

In general, we use the following notation.

*This says that the values of f(x) approach L as x*

*approaches a. In other words, the values of f(x) tend to get *
*closer and closer to the number L as x gets closer and *

## 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, we *
*never consider 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.*

## Example 1

Guess the value of Solution:

*Notice that the function f(x) = (x – 1)/(x*^{2} – 1) is not defined
*when x = 1, 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.*

*Example 1 – Solution*

*The tables below give values of f(x) (correct to six decimal *
*places) for values of x that approach 1 (but are not equal to *
1).

On the basis of the values in the tables, we make the guess that

cont’d

## The Limit of a Function

*Example 1 is illustrated by the graph of f in Figure 3. *

*Now let’s change f slightly by giving it the value 2 when*
*x = 1 and calling the resulting function g:*

**Figure 3**

## The Limit of a Function

*This new function g still has the same limit as x approaches *
1. (See Figure 4.)

## One-Sided Limits

## Example 6

*The Heaviside function H is defined by*

[This function is named after the electrical engineer Oliver
Heaviside (1850–1925) and can be used to describe an
*electric current that is switched on at time t = 0.] Its graph*
is shown in Figure 8.

## Example 6

*As t approaches 0 from the left, H(t) approaches 0. As t*

*approaches 0 from the right, H(t) approaches 1. There is no *
*single number that H(t) approaches as t approaches 0.*

Therefore does not exist.

cont’d

## One-Sided Limits

*We noticed in Example 6 that H(t) approaches 0 as t *
*approaches 0 from the left and H(t) approaches 1 as t*
approaches 0 from the right.

We indicate this situation symbolically by writing and

## One-Sided Limits

*The notation t* → 0^{–} indicates that we consider only values
*of t that are less than 0. *

*Likewise, t* → 0^{+} *indicates that we consider only values of t *
that are greater than 0.

## One-Sided Limits

Notice that Definition 2 differs from Definition 1 only in that
*we require x to be less than a.*

## One-Sided Limits

*Similarly, if we require that x be greater than a, we get “the *
**right-hand limit of f (x) as x approaches a is equal to L” **

and we write

*Thus the notation x* *→ a*^{+} *means that we consider only x *
*greater than a. These definitions are illustrated in Figure 9.*

**Figure 9**

## One-Sided Limits

By comparing Definition 1 with the definitions of one-sided limits, we see that the following is true.

## Example 7

*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 7 – 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.*

*Example 7 – Solution*

The graph also shows that and

(f) This time the left and right limits are the same and so, by (3), we have

*Despite this fact, notice that g(5)* ≠ 2.

cont’d

## Infinite Limits

## Infinite Limits

Another notation for lim_{x→a}*f(x) = ∞ is*
*f(x) → ∞* *as x* *→ a*

## Infinite Limits

Again, the symbol ∞ is not a number, but the expression
lim_{x→a}*f(x) = ∞ is often read as*

*“the limit of f(x), as x approaches a, is infinity”*

or *“f(x) becomes infinite as x approaches a”*

or *“f(x) increases without bound as x approaches a”*

## Infinite Limits

This definition is illustrated graphically in Figure 12.

**Figure 12**

## Infinite Limits

A similar sort of limit, for functions that become large

*negative as x gets close to a, is defined in Definition 5 and *
is illustrated in Figure 13.

## Infinite Limits

The symbol lim_{x→a}*f(x) = –∞ can be read as “the limit of *
*f(x), as x approaches a, is negative infinity” or “f(x) *

*decreases without bound as x approaches a.” As an *
example we have

## Infinite Limits

Similar definitions can be given for the one-sided infinite limits

*remembering that x → a*^{–} means that we consider only

*values of x that are less than a, and similarly x → a*^{+} means
*that we consider only x > a.*

## Infinite Limits

Illustrations of these four cases are given in Figure 14.

## Infinite Limits

## Example 10

*Find the vertical asymptotes of f(x) = tan x.*

Solution:

Because

*there are potential vertical asymptotes where cos x = 0. *

*In fact, since cos x* → 0^{+} *as x* → (π/2)^{–} *and cos x* → 0^{–} as
*x* → (π/2)^{+}*, whereas sin x is positive (near 1) when x is near *
π/2, we have

*Example 10 – Solution*

*This shows that the line x = *π/2 is a vertical asymptote.

*Similar reasoning shows that the lines x = *π*/2 + n*π, where
*n is an integer, are all vertical asymptotes of f(x) = tan x.*

The graph in Figure 16 confirms this.

cont’d