Answer:
[tex]\lim_{x\to 3^-} f(x)=-1[/tex]
Step-by-step explanation:
We have a piesewise function composed of three pieces. two line segments and one point.
The first line cuts at point (0,0) and ends at point (3, -1)
If we use these two points we can find the equation of the line.
The slope m is:
[tex]m = \frac{y_2-y_1}{x_2-x_1}\\\\m = \frac{-1 - (0)}{3-0}\\\\m = -\frac{1}{3}[/tex]
So the equation is:
[tex]y = -\frac{1}{3}x + b[/tex]
As the line cuts in (0,0) then [tex]b = 0[/tex] and the equation is:
[tex]y = -\frac{1}{3}x[/tex]
The equation of the second line is:
[tex]y = -4[/tex]
Now we can find f(x) (although it is not necessary to find the equation of f(x) because we have its graph)
[tex]f(x) = -\frac{1}{3}x[/tex] if [tex]x<3[/tex]; [tex]f(x)=-4[/tex] if [tex]x> 3[/tex]; [tex]f(x)= 7[/tex] if [tex]x = 3[/tex].
The limit of f(x) when x tends to 3 from the left [tex]\lim_{x\to 3^-} f(x)[/tex] is the limit of the function when x approaches 3 from the left. If x approaches 3 from the left then [tex]x <3[/tex]. If [tex]x <3[/tex] then f(x) is given by the line [tex]y = -\frac{1}{3}x[/tex] . Then the limit is -1 as seen in the graph
