Question

How do you find the limit as x approaches 2 of the piecewise function `f(x)={(4-x^2,if x<=2),(x-1,if x>2):}` ?

Answer 1

Since the left-hand limit and the right-hand limit at `x=2` do not match, the limit `lim_{x to 2}f(x)` does not exist.

Let us find the left-hand limit.
`lim_{x to 2^-}f(x)=lim_{x to 2^-}(4-x^2)=4-(2)^2=0`

Let us find the right-hand limit.
`lim_{x to 2^+}f(x)=lim_{x to 2^+}(x-1)=(2)-1=1`

Since the above one-sided limits at `x=2` do not match, the limit at `x=2` does not exist.