Question

How do you the limit of a piecewise function?

Answer 1

If you are looking for the limit of a piecewise defined function at the point where the function changes its formula, then you will have to take one-sided limits separately since different formulas will apply depending on which side you are approaching from. Here is an example.

For the following piecewise defined function

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

let us find the following limits.

(a) `lim_{x to 1}f(x)`

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

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

Since both limits give 1, `lim_{x to 1}f(x)=1`

(b) `lim_{x to 2}f(x)`

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

`lim_{x to 2^+}f(x)=lim_{x to 2^+}(2x-1)=2(2)-1=3`

Since the limits above are different, `lim_{x to 2}f(x)` does not exist.