Examples and further techniques

Through the examples in this page we show some of the most common techniques used in the calculation of limits.

1. . Use this technique when searching the limit of a rational function with x→±∞.

2. .

3. . Change the variable from x to t=1/x. As x tends to zero, t tends to infinity. Thus  .

4. . 

5. . Change the variable from x to e^x-1=t. As x tends to zero, t tends also to zero. The limit transforms in the reciprocal of limit 4: .

6. . This technique often works in the case of a rational function in the form 0/0.

7. .

8. .

9. . Here we have implicitly used a substitution x²=t. As x tends to zero, also t tends to zero. We can treat many other cases in the same way: usually the substitution will not be written explicitly. See the following example.

10. .