Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
Function Repository Resource:
Find the limiting value of an expression numerically
ResourceFunction["NLimit"][expr,z→z_{0}] numerically finds the limiting value of expr as z approaches z_{0}. 
WorkingPrecision  MachinePrecision  precision to use in internal computations 
Direction  Automatic  vector giving the direction of approach 
"Scale"  1  initial step size in the sequence of steps 
"Terms"  7  number of terms used to evaluate the limit 
Method  "EulerSum"  the method used to evaluate the result 
"WynnDegree"  1  degree used in Wynn's epsilon algorithm 
"EulerSum"  converts sequence to a sum 
"SequenceLimit"  constructs a sequence 
Find the limit at zero:
In[1]:= 

Out[1]= 

Find the limit at infinity:
In[2]:= 

Out[2]= 

The expression can be manifestly complex:
In[3]:= 

Out[3]= 

The limit point can be complex:
In[4]:= 

Out[4]= 

Expressions that approach their limiting value exponentially need fewer terms:
In[5]:= 

Out[5]= 

In[6]:= 

Out[6]= 

Increasing the number of terms can improve accuracy:
In[7]:= 

Out[7]= 

Error in numerical approximation:
In[8]:= 

Out[8]= 

Use more terms to reduce error:
In[9]:= 

Out[9]= 

Use "Scale" to avoid regions where the expression is undefined:
In[10]:= 

In[11]:= 

Out[11]= 

The function f(x) diverges for f(1), so choose the initial step to avoid this divergence:
In[12]:= 

Out[12]= 

Approach 0 along the negative real axis:
In[13]:= 

Out[13]= 

Approach 0 along the positive imaginary axis:
In[14]:= 

Out[14]= 

Approach 0 from the 3^{rd} quadrant, at 225°:
In[15]:= 

Out[15]= 

An example where the default method works fairly well:
In[16]:= 

Out[16]= 

Using "SequenceLimit" produces poorer results:
In[17]:= 

Out[17]= 

An example where the default method works poorly:
In[18]:= 

Out[18]= 

Here, "SequenceLimit" produces the correct result:
In[19]:= 

Out[19]= 

When using Method→"SequenceLimit", increasing "WynnDegree" may improve the accuracy of the limit:
In[20]:= 

Out[20]= 

Error with "WynnDegree"→1:
In[21]:= 

Out[21]= 

Error with "WynnDegree"→3:
In[22]:= 

Out[22]= 

Increasing WorkingPrecision alone does not produce a more accurate result:
In[23]:= 

Out[23]= 

Error with WorkingPrecision→20:
In[24]:= 

Out[24]= 

Error with WorkingPrecision→30:
In[25]:= 

Out[25]= 

To improve accuracy, the number of terms needs to be increased:
In[26]:= 

Out[26]= 

Find the limit of a numerically defined function:
In[27]:= 

In[28]:= 

Out[28]= 

Limits where parts of the expression have essential singularities:
In[29]:= 

Out[29]= 

In this case, the exact limit can be found:
In[30]:= 

Out[30]= 

Check:
In[31]:= 

Out[31]= 

Limits whose value approaches infinity are sometimes unable to be computed:
In[32]:= 

Out[32]= 

Wolfram Language 11.3 (March 2018) or above
This work is licensed under a Creative Commons Attribution 4.0 International License