It's best to sketch the graph first:
(i)
Why it's 1:
The negative - sign above the 1 on x->1- says we are moving along
the graph on the LEFT side of where x=1.
We move along the graph from LEFT to RIGHT, as the
x-coordinates left of 1 get closer and closer to 1, the
y-coordinates get closer and closer to the point (1,1),
which has y-coordinate 1. Even though the y-coordinates
don't reach 1, (since there is an open circle there),
they still approach 1, so the limit is 1 as x approaches
1 from the left side.
(ii)
Why it's also 1:
The + sign above the 1 on x->1+ says we are moving along
the graph on the RIGHT side of where x=1.
We move along the graph from RIGHT to LEFT, as the
x-coordinates right of 1 get closer and closer to 1, the
y-coordinates get closer and closer to the point (1,1),
which has y-coordinate 1. Even though the y-coordinates
don't reach 1, (since there is an open circle there),
they still approach 1, so the limit is 1 as x approaches
1 from the right side.
(iii) g(1) = 5
Why it's 5:
Although the graph approaches the point (1,1) from both
the left and the right side it doesn't get there, because
as soon as it gets almost there the graph jumps way up to
(1,5), and g(1) = 5 just means that the graph contains the
point (1,5).
(iv)
Why it's -2:
The negative - sign above the -2 on x->2- says we are moving along
the graph on the LEFT side of where x=2.
We move along the graph from LEFT to RIGHT, as the
x-coordinates left of 2 get closer and closer to 2, the
y-coordinates get closer and closer to the point (2,-2),
which has y-coordinate -2. In this case the y-coordinates
actually do reach -2, (since there is an CLOSED circle there),
they approach (and reach) -2, so the limit is -2 as x approaches
2 from the left side.
(v)
Why it's 0:
The + sign above the 2 on x->2+ says we are moving along
the graph on the RIGHT side of where x=1.
We move along the graph from RIGHT to LEFT, as the
x-coordinates right of 2 get closer and closer to 2, the
y-coordinates get closer and closer to the point (2,0),
which has y-coordinate 0. Even though the y-coordinates
don't reach 0, (since there is an open circle there),
they still approach 0, so the limit is 0 as x approaches
2 from the right side.
(vi)
This DOES NOT EXIST.
Why it doesn't exist:
Notice that there is neither sign + or - above 2 on x->2.
Since there neither sign is written above what the limit
of x approaches, it means that the graph approaches the
same limit on BOTH SIDES of 2. But this is clearly false.
It (the y-coordinates) approaches -2 on the left side of
x=2 and approaches 0 on the right side of x=2.
You weren't asked, but you should have been:
(vii)
Why it's 1:
There is no + or - sign above the 1 in "x->1", so that
means the graph (the y-coordinates) approach the same number
1 from both of x=1.
Edwin
