Question

Can someone answer this?

Answer 1

`\lim_(x\to 3^-) f(x)=-1`

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:

`m = (y_2-y_1)/(x_2-x_1)\\\\m = (-1 - (0))/(3-0)\\\\m = -(1)/(3)`

So the equation is:

`y = -(1)/(3)x + b`

As the line cuts in (0,0) then
`b = 0` and the equation is:

`y = -(1)/(3)x`

The equation of the second line is:

`y = -4`

Now we can find f(x) (although it is not necessary to find the equation of f(x) because we have its graph)

`f(x) = -(1)/(3)x` if
`x<3`;
`f(x)=-4` if
`x> 3`;
`f(x)= 7` if
`x = 3`.

The limit of f(x) when x tends to 3 from the left
`\lim_(x\to 3^-) f(x)` is the limit of the function when x approaches 3 from the left. If x approaches 3 from the left then
`x <3`. If
`x <3` then f(x) is given by the line
`y = -(1)/(3)x` . Then the limit is -1 as seen in the graph