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)

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ishpiro | College Teacher | (Level 1) Educator

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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.

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