For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.
d) h(-3) is undefined (the point (-3, 4) is bubbled out, and there is no other point anywhere at x =-3)
e) `lim_(x>0-) h(x) = 1` (the left branch of the graphed function is approaching y = 1 when x is approaching 0 from the left.)
f) `lim_(x->0+) h(x)= -1` (the right branch of the graphed function is approaching
y = -1 when x is approaching 0 from the right.)
g) `lim_(x->0) h(x)` does not exist because the right-sided and left-sided limits are not equal, as shown above.
h) h(0) = 1 (the solid dot for x = 0 is at (0, 1))
i) `lim_(x->2) h(x) = 2` because the function is approaching y = 2 when x approaches x = 2 from either left or right.
j) h(2) is undefined (there is no solid dot anywhere at x = 2)
k) `lim_(x->5+) h(x)` Is not really clear from this graph, but it appears to me that it is 3.
l) `lim_(x->5-) h(x) ` does not exist, because as x approaches 5 from the left, the function does not approach any particular y value, but keeps going up and down infinitely.
We are asked to find the limit of h(x) as x approaches -3 from the left, the right, and the limit if it exists.
The limit as x approaches -3 from the left is 4.
The limit as x approaches -3 from the right is 4.
The limit as x approaches -3 is 4.
Note that h(x) is not defined at x=-3. This does not matter for the limit. We are only concerned with the behavior of the function in a neighborhood of -3.
No matter how small a real number epsilon>0 you choose, the function will be less than epsilon away from 4 for some neighborhood of -3.