Limits of a piecewise defined function?

Question

I have the function:

h(x)={x if x<0

x^2 if 0<x<2

8-x if x>2

I have to find the limit as x approaches 0, but since 0 doesn't fit into any the fuctions, does it not exsist or what?

Thank you in advanced!

Answer 1

h(0) does not have to exist for the limit to exist. The limit is simply the value h(x) gets close to from both the left and right. In this case h(x) approaches 0 from the left and the right. 0 is the limit  

Answer 2

The limit exists, even though the point where x=0 does not. If you graph it out, as x approaches zero, the y appears to &quot;get close&quot; to 0 as well. As 0 happens to be a singular, finite number, your limit does indeed exist.

The situations that limits do not exist are when the limit oscillates between more than one value (e.g. the limit of sin(x)) or when it approaches infinity.  

Answer 3

limit of f(x) as x---&gt; a exists if lim {from the left of x=a} of f(x) = lim{ from the right of x = a} f(x)....the value x = a need not be in the domain of f....Given an ε find a δ so that if 0 &lt; | x - a | &lt; δ then |f(x) - L | &lt; ε, notice that if x = a then the 1st inequality is not met