Proving continuity

Question

F(x)={2x+5; x<2.           A) lim x~2-

         {9; x=2.                 B) lim x~2+

         {x^3; x>2.             C) is f(x) continuous at x=2

I completely forget how to solve this.

are 9, 10, and no correct?

Answer 1

Kim,

 

You're almost right; I think you just made one small calculation error.

 

Your answer for (A) is correct. However, for (B), 2³ = 8, not 10. Your reasoning still holds for (C): since (A) and (B) are not equal, the function can't be continuous at x = 2.

Answer 2

We have a piecewise function where each part of a function has its own domain.  A function is continuous under these 3 conditions. NO EXCEPTIONS!

 

1)    lim F(x)

       x -->c          from the left

 

 

2)    lim F(x)

       x --> c⁺    from the right

 

 

3)   F(x) is the value at c.

 

 

In this situation, c = 2.   You will need to graph each part of the piecewise function to see if F(x) is continuous at c=2.