Choose one of the following topics:
The table of values we found earlier for x nearing 2 was:
f(1.5)  =  2.5  f(2.5)  =  3.5 
f(1.7)  =  2.7  f(2.3)  =  3.3 
f(1.8)  =  2.8  f(2.2)  =  3.2 
f(1.9)  =  2.9  f(2.1)  =  3.1 
f(1.99)  =  2.99  f(2.01)  =  3.01 
f(1.999)  =  2.999  f(2.001)  =  3.001 
The values certainly look like they are approaching 3, but how do we know for certain? Perhaps they are approaching 3.000075 or 2.999996. What we need is a precise definition of a limit, which will tell us when we are exactly correct.
Let f be
a function defined on some open interval that contains the number a,
except possibly at a itself. We say the limit of
f(x) as x approaches a is L, and
we write
if for every number > 0 there is a corresponding number > 0 such that
whenever . 
Using our example function, f(x) is as above, a=2, and we think that L=3. Let be any positive number. Then we can choose to be equal to . In which case, if 0 < x2 < ,
We can cancel the (x3) term from the top and bottom of the fraction, as well as one of the (x2) terms from the top with the same on the bottom, to get
.
So, since the definition fits exactly, we can state with certainty that the limit as x approaches 2 of f(x) is 3. Also, we can notice that if we used any value other than 3 for L, we wouldn't get the cancellation we did in the last step, so we would not have been able to fit the definition.
Below are the definitions of several related limit concepts.
Let f be
a function defined on some open interval (b,a) [or
(a,b)]. We say the lefthand [or righthand] limit of
f(x) as x approaches a is L, (or
the limit of f(x) as x approaches a from
the left [or right] is L) and
we write
if for every number > 0 there is a corresponding number > 0 such that
whenever

Let f be
a function defined on some open interval that contains the number a,
except possibly at a itself. We say the limit of
f(x) as x approaches a is [negative]
infinity, and we write
if for every number N > 0 [N < 0] there is a corresponding number > 0 such that f(x) > N [f(x) < N] whenever . 
Let f be
a function defined on some open interval from a to infinity [from
negative infinity to a]. Then we say the limit of
f(x) as x approaches [negative] infinity is
L, and we write
if for every number > 0 there is a corresponding number N such that
whenever x > N [x < N]. 
Let f be
a function defined on some open interval from a to infinity {from
negative infinity to a}. We say the limit of
f(x) as x approaches {negative} infinity is [negative]
infinity, and we write
if for every number M > 0 there is a corresponding number N such that f(x) > M [f(x) < M] whenever x > N {x < N}. 
Similar definitions can be made for infinitevalued left or righthand limits.
Choose one of the following topics:
If you have questions or comments, don't hestitate to contact us.