The important thing to note here is that a limit is defined when the value of f(x) as it approaches x is the same when approaching from both the left and the right sides (unless a specific side from which to approach has been defined in the question, eg. lim (x -> 10⁻) f(x) or lim (x -> 10⁺) f(x) ).
1. The first line states that the limit of f(x) as x approaches 10 is 0. This means that the value of f(x) as x approaches 10 from both the left and the right sides must be 0.
Looking at the graph, we can see that this is indeed the case, and may write this as:
lim (x -> 10⁻) f(x) = lim (x -> 10⁺) f(x) = 0
Thus, the statement is True.
2. The second line states that the limit of f(x) as x approaches -2 from the right side is 3.
Looking at the graph, whilst f(x) approaches 3 as x approaches -2 from the left side, f(x) approaches 7 as x approaches -2 from the right side. We can write this as:
lim (x -> -2⁺) f(x) = 7
Thus, since lim (x -> -2⁺) f(x) ≠ 3, the statement is False.
3. The third line states that the limit of f(x) as x approaches -8 is the value of f(x) at x = -8 (ie. the value of f(x) for which the graph exists at x = -8).
Looking at the graph, we can see that the graph of f(x) approaches -6 as x approaches -8 from both the left and right sides. We can write this as:
lim (x -> -8⁻) f(x) = lim (x -> -8⁺) f(x) = -6
Now, this is where a knowledge of what open and closed circles represent on a graph is crucial. A closed circle means that a point exists on a graph, whereas an open circle means that there is a break in the graph at that point.
If we look at the graph, we can see that the closed circle at x = -8 is actually the value f(x) = -3. We can write this as:
f(-8) = -3
Moving from the line before this, where we defined the limit as -6, we can thus write:
lim (x -> -8) f(x) ≠ -3
Therefor, lim (x -> -8) f(x) ≠ f(-8)
Thus, the statement is False.
4. The fourth line states that the limit of f(x) as x approaches 6 is 5. Thus, for this to be true, f(x) must approach 5 as x approaches 6 from both the left and right sides.
Looking at the graph, we can see that as x approaches 6 from the left side, f(x) approaches 2; whilst as x approaches 6 from the right side, f(x) approaches 5. We can write this as:
lim (x -> 6⁻) f(x) = 2
lim (x -> 6⁺) f(x) = 5
These limits are not the same:
lim (x -> 6⁻) ≠ lim (x -> 6⁺), therefor a limit does not exist at x = 6.
Thus, this statement is False.
Hope this helps, but if anything is unclear, please feel free to leave a comment below :)
