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Summation of SeriesResidues

3.6.8 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.

In[1]:= Cos[x^2] /. x -> 0

Out[1]=

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.

Finding limits.

This gives the correct value for the limit of as .

In[2]:= Limit[ Sin[x]/x, x -> 0 ]

Out[2]=

No finite limit exists in this case.

In[3]:= Limit[ Sin[x]/x^2, x -> 0 ]

Out[3]=

Limit can find this limit, even though you cannot get an ordinary power series for at .

In[4]:= Limit[ x Log[x], x -> 0 ]

Out[4]=

The same is true here.

In[5]:= Limit[ ( 1 + 2 x ) ^ (1/x), x -> 0 ]

Out[5]=

The value of Sign[x] at x=0 is 0.

In[6]:= Sign[0]

Out[6]=

Its limit, however, is 1. The limit is by default taken from above.

In[7]:= Limit[Sign[x], x -> 0]

Out[7]=

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 1. Limit represents values with bounded variation using Interval objects. In general, Interval[xmin, xmax] represents an uncertain value which lies somewhere in the interval xmin to xmax.

Limit returns an Interval object, representing the range of possible values of near its essential singularity at .

In[8]:= Limit[ Sin[1/x], x -> 0 ]

Out[8]=

Mathematica can do arithmetic with Interval objects.

In[9]:= (1 + %)^3

Out[9]=

Mathematica represents this limit symbolically in terms of an Interval object.

In[10]:= Limit[ Exp[Sin[x]], x -> Infinity ]

Out[10]=

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.

Directional limits.

The function has a different limiting value at , depending on whether you approach from above or below.

In[11]:= Plot[1/x, {x, -1, 1}]

Out[11]=

Approaching from below gives a limiting value of .

In[12]:= Limit[ 1/x, x -> 0, Direction -> 1 ]

Out[12]=

Approaching from above gives a limiting value of .

In[13]:= Limit[ 1/x, x -> 0, Direction -> -1 ]

Out[13]=

Limit makes no assumptions about functions like f[x] 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 f, so it leaves this limit unevaluated.

In[14]:= Limit[ x f[x], x -> 0 ]

Out[14]=Limit[x f[x], x 0]

Summation of SeriesResidues