One of the main reasons for discussing limits of functions is to allow a definition of the derivative. Continuity, derivatives, and integrals are the three main topics in calculus. So far, all that has been discussed is continuity. The derivative will be in the next chapter.
In this section, functions will be defined on some subset of ℝ.
Definition 4.11.1 Let f be a function which is defined on D

if and only if the following condition holds. For all ε > 0 there exists δ > 0 such that if

then,

If everything is the same as the above, except y is required to be larger than x and f is only required to be defined on

If f is only required to be defined on

Limits are also taken as a variable “approaches” infinity. Of course nothing is “close” to infinity and so this requires a slightly different definition.

if for every ε > 0 there exists l such that whenever x > l,
 (4.4) 
and

if for every ε > 0 there exists l such that whenever x < l, 4.4 holds.
The following pictures illustrate some of these definitions.
In the left picture is shown the graph of a function. Note the value of the function at x equals c while lim_{y→x+}f
Proof:Let ε > 0 be given. There exists δ > 0 such that if 0 <

Therefore, for such y,

Since ε > 0 was arbitrary, this shows L = L_{1}. The last claims are similar. For example, if it is lim_{y→∞}, then there exists l such that if y > l then

and then the same argument just used shows that
Another concept is that of a function having either ∞ or −∞ as a limit. In this case, the values of the function do not ever get close to their “limit” because nothing can be close to ±∞. Roughly speaking, the limit of the function equals ∞ if the values of the function are ultimately larger than any given number. More precisely:
Definition 4.11.3 If f
It may seem there is a lot to memorize here. In fact, this is not so because all the definitions are intuitive when you understand them. Everything becomes much easier when you understand the definitions. This is usually the way it works in mathematics.
In the following theorem it is assumed the domains of the functions are such that the various limits make sense. Thus, if lim_{y→x} is used, it is to be understood the function is defined on
Theorem 4.11.4 In this theorem, the symbol lim_{y→x} denotes any of the limits described above. Suppose lim_{y→x}f
 (4.5) 
 (4.6) 
and if K≠0,
 (4.7) 
Also, if h is a continuous function defined in some interval containing L, then
 (4.8) 
Suppose f is real valued and lim_{y→x}f
Proof:The proof of 4.5 is left for you. It is like a corresponding theorem for continuous functions. Next consider 4.6. Let ε > 0 be given. Then by the triangle inequality,
≤ +


≤ +
.  (4.9) 
There exists δ_{1} such that if 0 <

and so for such y, and the triangle inequality,
 (4.10) 
Now let 0 < δ_{2} be such that for 0 <

Then letting 0 < δ ≤ min

and this proves 4.6. Limits as x →±∞ and one sided limits are handled similarly.
The proof of 4.7 is left to you. It is just like the theorem about the quotient of continuous functions being continuous provided the function in the denominator is non zero at the point of interest.
Consider 4.8. Since h is continuous at L, it follows that for ε > 0 given, there exists η > 0 such that if

Now since lim_{y→x}f

Therefore, if 0 <

The same theorem holds for one sided limits and limits as the variable moves toward ±∞. The proofs are left to you. They are minor modifications of the above.
It only remains to verify the last assertion. Assume f
A very useful theorem for finding limits is called the squeezing theorem.
Proof: If L ≥ h

If L < h

Therefore,

Now let ε > 0 be given. There exists δ_{1} such that if 0 <

and there exists δ_{2} such that if 0 <

Letting 0 < δ ≤ min
≤ +


< ε∕2 + ε∕2 = ε.■ 
Theorem 4.11.6 For f : I → ℝ, and I is an interval of the form
Proof: You fill in the details. Compare the definition of continuous and the definition of the limit just given. ■
Example 4.11.7 Find lim_{x→3}
Note that

It follows from the definition that this limit equals 6.
You should be careful to note that in the definition of limit, the variable never equals the thing it is getting close to. In this example, x is never equal to 3. This is very significant because, in interesting limits, the function whose limit is being taken will not be defined at the point of interest. The habit students acquire of plugging in the point to take the limit is only good on useless and uninteresting limits which are not good for anything other than to give a busy work exercise.
Example 4.11.8 Let

How should f be defined at x = 3 so that the resulting function will be continuous there?
The limit of this function equals 6 because for x≠3,

Therefore, by Theorem 4.11.6 it is necessary to define f
Example 4.11.9 Find lim_{x→∞}
Write

Example 4.11.10 Show lim_{x→a}
There are two cases. First consider the case when a > 0. Let ε > 0 be given. Multiply and divide by

Now let 0 < δ_{1} < a∕2. Then if
=
≤ 

≤
. 
Now let 0 < δ ≤ min

Next consider the case where a = 0. In this case, let ε > 0 and let δ = ε^{2}. Then if 0 < x − 0 < δ = ε^{2}, it follows that 0 ≤