Graphing and finding the limit

Question

 Graph F(x)= -x^2+2, for x doesn't = 1 

          0,        x =1

 

 

A) Find lim  f(x)

           x>1

 

B) Find f(1)

 

C is f continuous at 1?

Answer 1

I see that the way the equation is defined means that it equals -x² + 2 when x ≠ 1, and 0 when x = 1.

 

You can graph that yourself, and you'll see that it looks like a parabola with a missing point at x = 1, and a point separate from the parabola at (1, 0).

 

The limit of -x² + 2 as x approaches 1 is the value of -x² + 2 at 1.  The fact that the function is defined to have a different value at x = 1 does not affect the limit, because when doing limits, the value of x never actually GETS to 1.  You're just considering the behavior of the function AROUND 1.

 

The function is NOT continuous at x = 1.  In order for it to be continuous, the limit as x approaches 1 has to be equal to the value of the function AT 1, which is not the case.