Limits of rational functions
         The limit of a rational function at infinity containing roots (irrational expressions)
         The limit of a rational function at a point containing irrational expressions, use of substitution
         Evaluating the limit of a rational function containing irrational expressions using rationalization
The limit of a rational function at infinity containing roots (irrational expressions)
We use the same method we used to evaluate the limit of a rational function at infinity that is, isolate and cancel a common factor of x from both the numerator and denominator and than find the limit of the equivalent expression.
Example:  Evaluate the limit  
Solution:  
Example:  Evaluate the limit  
Solution:  
The limit of a rational function at a point containing irrational expressions use of substitution
Use of the method of substitution to avoid the indeterminate form of an expression.
Example:  Evaluate the limit  
Solution:     Let substitute,   x + 1 =  y6,   then as  x 0  then   y 1,   therefore    
 
Example:  Evaluate the limit  
Solution:     Let substitute,   xy12,   then as  x 1  then   y 1,   therefore
 
Evaluating the limit of a rational function containing irrational expressions using rationalization 
To avoid the indeterminate form of the irrational expression we rationalize the numerator or the denominator as appropriate.
Example:  Evaluate the limit  
Solution:     Let rationalize the numerator,
 
Example:  Evaluate the limit  
Solution:     Let rationalize both the numerator and denominator,
 
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