Rational function is the ratio of two polynomial functions and it is usually represented as
where g(x) ≠ 0.
Consider the following limit of a rational function.
When you evaluate the above the limit of a rational function, you may get one of the following results.
Case 1 :
The above limit exists and it is equal to the finite value received in the last step of evaluation.
Case 2 :
Dividing a non-zero number by zero is undefined. So, the above limit does not exists.
Case 3 :
Dividing zero by zero is called indeterminate form (not undefined). Indeterminate form is not a final answer for the limit of a rational function. We can evaluate such limit by simiplifying the given rational function.
The following steps will be useful to evaluate the limit of a rational function which initially results the inderterninate form 'zero by zero'.
Step 1 :
Factor the polynomials in numerartor and denominator.
Step 2 :
Cancel out the common factors found in numerator and denominator (simplification).
Step 3 :
Substitute the given limit for the variable x and evaluate. The answer will be a finite value.
Evaluate the following limits :
Problem 1 :
Solution :
Problem 2 :
Solution :
Since the evaluation of the given limit is undefined, the limit does not exist.
Problem 3 :
Solution :
Problem 4 :
Solution :
Since the evaluation of the given limit results indeterminate form 'zero by zero', simplify the rational function and substitute the given limit for x and evaluate.
Note :
When you evaluate limits of rational functions, it is advisable to simplify the rational function first (if possible), then substitute the given limit for the variable.
Problem 5 :
Solution :
Use the following algebraic identity and factor the expression in denominator.
a2 - b2 = (a + b)(a - b)
Problem 6 :
Solution :
Use the following algebraic identity and factor the expression in denominator.
a3 + b3 = (a + b)(a2 - ab + b2)
Problem 7 :
Solution :
Problem 8 :
Solution :
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