Limits are the foundation of calculus. As we have mentioned in our section on history, Jean d'Alembert defined the derivative, dy/dx, using limits. Here, we give a brief explanation on limits, and then proceed to show how the derivative is obtained, by first principles.

 

The limit is written as

                                          

which means the value that f(x) approaches as x tends to a. The limit exists only if it has a defined value. If the calculated limit approaches infinity, it does not exist.

 

Results and Theorems

 

Basic Results

 

Performing Operations on Limits

 

Limits Involving Infinity

 

 

Examples: Evaluate the following limits

                              

                       

 

Solutions: