Question

How do you find one sided limits and limits of piecewise functions?

Answer 1

The only thing you need to worry about is that you choose the right formula since piecewise defined functions have multiple formulas.

Let us look at an example.

`f(x)={(x+2 if x<-2),(x^2 if -2 le x <1),(1/x if 1 le x):}`

(1) `lim_{x to -2^-}f(x)=lim_{x to -2^-}(x+2)=-2+2=0`

(2) `lim_{x to -2^+}f(x)=lim_{x to -2^+}x^2=(-2)^2=4`

(3) `lim_{x to -2}f(x)` does not exist since `(1) ne (2)`

(4) `lim_{x to 1^-}f(x)=lim_{x to 1^-}x^2=(1)^2=1`

(5) `lim_{x to 1^+}f(x)=lim_{x to 1^+}1/x=1/1=1`

(6) `lim_{x to 1}f(x)=1` since (`(4)=(5)`)

As you can see above, you simply need to choose the correct formula depending on which way it is approaching from. Two-sided limit exists only when the left-hand limit and the right-hand limit are the same.