We now have to make precise, mathematical sense out of the statement

lim x® a  f(x) = L.

Recall the informal definition:

Definition 1 We say

lim x® a  f(x) = L

if we can make f(x) as close as we need to L , just by taking x close enough to a .

In order to make that precise, we need first to figure out how to quantify how close f(x) is to L . The distance between them is |f(x)-L| . So, how close do we need it? Say within e, which is some positive number. (All calculus texts always use the letter e for this.) So, we need to be able to arrange |f(x)-L| < e, just by taking x close enough to a . The distance from x to a is |x-a| . Close enough should be some (small) positive distance d. So, the definition above becomes:

Definition 2 Let f(x) be defined in a neighborhood of a , except perhaps at a itself. That is, assume for some c < a and b > a , f is defined on the intervals (c,a) and (a,b) . Then, we say that

lim x® a  f(x) = L

if, for every e > 0 , there is a d > 0 so that, if 0 < |x-a| < d, then |f(x)-L| < e.

Note 4 Notice how I snuck that 0 < |x-a| in there. That just means that x ¹ a . Remember, in limits we don't care how, or if, f(a) is defined, we want to know what f(x) is tending towards as x® a . We want to know how f(a) should be defined.

One-sided limits are handled the same way, except that 0 < |x-a| < d is replaced by a < x < a+d for the right-hand limit

lim x® a+  f(x),

or a-d < x < a for the left-hand limit

lim x® a+  f(x).

1.5.1  Precise definition of limits at ¥

To begin with, recall the informal definition of the situation

lim x® ¥  f(x) = ¥.

In that case, we said informally that ``as x gets larger and larger, so does f(x) ''. What does that mean? It means that, no matter how large you insist that f(x) be, I can arrange that just by taking x large enough.

Now. let's make it precise. How large do you want me to make f(x) ? It would be some number, E . I have to make f(x) larger than that, at least. But, I'm only allowed to do that by taking x large enough, say x > D . If the limit

lim x® ¥  f(x) = ¥

is true, I have to be able to do this, no matter what. Here goes:

Definition 5 Assume that, for some number c , f(x) is defined for all x > c . Then, we say that

lim x® ¥  f(x) = ¥

if, for each number E , there is a number D so that, if x > D , then f(x) > E .

1.5.2  Finite limits as x® ¥

A more subtle case is the informal statement of the condition that

lim x® ¥  f(x) = L.

There, we say informally that ``in order to make f(x) as close as we need to L , we just need to take x large enough''.

Assume that, for some number c , f(x) is defined for all x > c . Then, we say that

lim x® ¥  f(x) = L

if, for each number e > 0 , there is a number D so that, if x > D , then |f(x)-L| < e.

1.5.3  Infinite limits as x® a

Definition 6 A function f(x) has an infinite limit as x approaches a (or has a vertical asymptote at x = a ),

lim x® a  f(x) = ¥,

if, for each E > 0 , there is a d > 0 so that whenever 0 < |x-a| < d, |f(x)| > E .

You can figure out what the analogue would be for one-sided limits, or for limits going specifically to +¥, -¥, or both ways, ±¥. Of course, this last case is really the same as the definition we just gave.

Example 1

Show that

lim x® 1  3x+5 = 8.

Example 2

Show that

lim x® 2  x2+3x = 10.

Example 3

Example 4

Show that

Example 5

Exercises

Exercise 1 Find a number d sufficiently small so that the distance from f(x) = 2x²+3x-1 to 4 is less than 1/100 if |x-1| < d.

d =

Exercise 2 Show that

lim x® ¥  (x/2+sinx) = ¥.

Answer

Exercise 3 Show that

lim x® ¥  x2+3x+2 x2-1 = 1.

Answer

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