In a series cutting experiments, Jimmy summarized the relationship between force to cut the material and the heat released from the material. The heat is a piecewise continuous function of force. He wants to analysis the heat released when force is 1 and 2 kN. The function is:


Since f(x) is a piecewise function, the functions for f(x) on the left and right hand side of 1 kN and 2 kN. The way to analysis the boundary heat conductive is to consider both side heat conductive at the boundary.

First, consider the case of x approaches 1-:



Second, consider the case of x approaches 1+:


Third, consider the case of x approaches 2-:


Finally, consider the case of x approaches 2+:



Piecewise f(x) function diagram
  • According to left and right hand limit,


    In this situation, although both left and right hand limit exist, the limit at 1 kN does not exist. The heat conductive function is not a continuous function at this point. The function has a jump. The same situation happens at 2 kN.

  • In order to solve the limit of a function, many limit laws can be applied. The proper application of the limit law can make the problem solving process easier. Of course, it does take practice.