How do I determine limits from a piece wise function?

Question

 

 

 How do you determine limits from a piece wise function in calculus?

Answer 1

For each point at which a limit is considered (say, x₀  for a function y = y(x)) you have to check for left-side and right-side limits:

 

                       lim y(x)   = a     and    lim y(x) = b

                       x→x₀ + 0                    x →x₀ -0 

 

If they are not equal (a ≠ b) the function is piece wise (the function is "jumping" from point "a" to point "b" up if a<b, or down if b< a). 

 

A piece wise function may have a form of a "broken line" if at some points its derivative doesn't exist (you don't know how to draw a tangent line to the graph of the function at these points because the slope of the tangent is undefined).).

Answer 2

looking at a graph is the easiest.

 

There is no limit at places where the pieces are not connected.

 

One sided limits may exist however.

 

There is a limit if the pieces meet where there is a hole and the function has no value there.

 

There are lots of possibilities.