Finding Limits
In doing many kinds of calculations, you need to evaluate expressions when variables take on particular values. In many cases, you can do this simply by applying transformation rules for the variables using the 
operator.
You can get the value of
at 0 just by explicitly replacing
with 0, and then evaluating the result.
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In some cases, however, you have to be more careful.
Consider, for example, finding the value of the expression 
when 
. If you simply replace 
by 
in this expression, you get the indeterminate result 
. To find the correct value of 
when 
, you need to take the limit.
| Limit[expr,x->x0] | find the limit of expr when x approaches  |
Finding limits.
This gives the correct value for the limit of
as
.
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No finite limit exists in this case.
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Limit can find this limit, even though you cannot get an ordinary power series for
at
.
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The value of
Sign[x] at
is
0.
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Its
limit, however, is
. The limit is by default taken from above.
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Not all functions have definite limits at particular points. For example, the function 
oscillates infinitely often near 
, so it has no definite limit there. Nevertheless, at least so long as 
remains real, the values of the function near 
always lie between 
and 
. Limit represents values with bounded variation using Interval objects. In general, Interval[{xmin
represents an uncertain value which lies somewhere in the interval 
to 
.
Limit returns an
Interval object, representing the range of possible values of
near its essential singularity at
.
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Mathematica can do arithmetic with
Interval objects.
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Mathematica represents this limit symbolically in terms of an
Interval object.
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Some functions may have different limits at particular points, depending on the direction from which you approach those points. You can use the Direction option for Limit to specify the direction you want.
| Limit[expr,x->x0,Direction->1] | find the limit as x approaches  from below |
| Limit[expr,x->x0,Direction->-1] | find the limit as x approaches  from above |
Directional limits.
The function
has a different limiting value at
, depending on whether you approach from above or below.
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Approaching from below gives a limiting value of
.
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Approaching from above gives a limiting value of
.
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Limit makes no assumptions about functions like 
about which it does not have definite knowledge. As a result, Limit remains unevaluated in most cases involving symbolic functions.
Limit has no definite knowledge about
, so it leaves this limit unevaluated.
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