Use the graph of f to state the value of each quantity, if it exists.  If it does not exist, explain why. a) `lim_(x->2-) f(x)` b) `lim_(x->2+) f(x)` c) `lim_(x->2) f(x)` d) f(2) e)...

Use the graph of f to state the value of each quantity, if it exists.  If it does not exist, explain why.

a) `lim_(x->2-) f(x)`

b) `lim_(x->2+) f(x)`

c) `lim_(x->2) f(x)`

d) f(2)

e) `lim_(x->4) f(x)`

f) f(4)

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Expert Answers
ishpiro eNotes educator| Certified Educator

a) `lim_(x ->2-) f(x)`

This notation means "the limit of function f(x) when x is approaching 2 from the left side." (The left side is indicated by minus after 2, which means that x remains SMALLER than 2 while approaching 2.)

As can be seen from the graph on the image, f(x) is approaching y = 3 when x approaches 2 from the left (look at the left branch of the graphed function.) So,

`lim_(x -> 2-) f(x) = 3`

b) `lim_(x->2+) f(x)`

Now we are looking for the limit of f(x) when x is approaching 2 form the right side. (The right side is indicated by plus after 2, which means that x remains LARGER than 2 while approaching 2.)

As can be seen from the graph on the image, f(x) is approaching y = 1 when x approaches 2 from the right (look at the right branch of the graphed function.) Note that the limit still exists, even though the point (2,1) is taken out (there is a hole at that point instead of a solid dot.) So,

`lim_(x->2+) f(x) = 1`

c) `lim_(x ->2) f(x)`

The limit of f(x) at point 2 does not exist because in order for the limit at a given point to exist, the right-sided and left-sided limits have to be equal. As shown in a) and b), the right-sided limit is 1 and the left sided limit is 3, so they are NOT equal. Thus,

`lim_(x->2) f(x) ` does not exist.

d) f(2)

This notation means "the value of function f(x) at x = 2." In other words, it is a y-coordinate of a point that has the x-coordinate 2. On the graph, the solid dot at the point which has x = 2 is at (2, 3). So, f(2) = 3.

e) `lim_(x->4) f(x) `

To find this limit, consider first the right-sided and left-sided limits at x = 4. As can be seen from the graph, f(x) approaches y = 4 when x aprroaches 4 from either left or right side. So the right-sided and left-sided limits are equal and equal 4. Therefore,

`lim_(x->4) = 4`

f) f(4)

There is NO solid dot at the point that would have been on the graph of f(x) with the x-coordinate x = 4. This means there is no y-coordinate, or value of y,  that corresponds to x = 4. So, f(4) is undefined.

justaguide eNotes educator| Certified Educator

The graph of the function that is given can be used to determine all the required values in the question.

1. `lim_(x->2-) f(x)`

`lim_a- f(x)` refers to the value of f(a) as x approaches a from values smaller than a. In the graph, it refers to the value of f(a) as we move toward "a" from the left hand side.

The graph of the function is not continuous at 2.

If we move toward 2 from the left hand side, the value of f(2) is 3. In the graph, there is a closed dot at f(2), and it lies at (2, 3), as we move from the left. Therefore `lim_(x->2-) f(x) = 3`

2. `lim_(x->2+) f(x)`

`lim_(x->a+) f(x)` refers to the value of f(a) as x approaches "a" from values greater than "a." In the graph, it refers to the value of f(a), as we move toward "a" from the right hand side. 

If we move toward x = 2 from the right hand side, there is an open dot at f(2). Though the value of f(2) in this part of the graph is not defined, the value of `lim_(x->2+) f(x)` can be determined. This is equal to 1.

f(2)The value of `lim_(x->2+)` is 1.

3. `lim_(x->2) f(x)`

The value of `lim(x->a)` can be determined if `lim_(x->a-) f(x) = lim_(x->a+)f(x)` .

In the graph of the function provided, we see that this is not the case. Therefore `lim_(x->) f(x)` is not defined.

4. f(2)

From the graph we see that the value of f(2) = 3

5. `lim_(x->4) f(x)`

In the graph at the point where x = 4, there is an open dot. At this point the value of f(x) is not defined. However, the value of `lim_(x->4) f(x)` can nevertheless be determined. This is the case because `lim_(x->a) f(x)` is not the value of the function at x = a, rather it is the value that f(x) takes as x becomes infinitesimally closer to a. 

The value of `lim_(x->4) f(x)` is 4.

6. The value of f(4) is not defined.

f(x) is not defined.