Question

Please explain how this function behaves when it approaches the given x values!

Answer 1

Answer and Explanation:

This is a piecewise function because it is defined by more than two functions. Basically, we want to take the limit here. Recall that if a function
`f(x)` approaches some value
`L` as
`x` approaches
`a` from both the right and the left, then the limit of
`f(x)` exists and equals
`L`. Here we won't calculate the limit, but apply some concepts of it. So:

a.
`as \ x \rightarrow +\infty, \ k(x) \rightarrow +\infty`

Move on the x-axis from the left to the right and you realize that as x increases y also increases without bound.

b.
`as \ x \rightarrow -\infty, \ k(x) \rightarrow 0`

Move on the x-axis from the right to the left and you realize that as x decreases to negative values y approaches zero.

c.
`as \ x \rightarrow 2, \ k(x) \rightarrow 0`

Since the function is continuous here, we can say that
`k(2)=0`

d.
`as \ x \rightarrow -2, \ k(x) \rightarrow 0`

The function is discontinuous here, but
`k(-2)` exists and equals 0 as the black hole indicates at
`x=-2`.

e.
`as \ x \rightarrow -4, \ k(x) \rightarrow 2`

The function is also discontinuous here, but the black hole indicates that this exists at
`x=-4`, so
`k(-4)=2`

f.
`as \ x \rightarrow 0, \ k(x) \rightarrow 4`

Since the function is continuous here, we can say that
`k(0)=4`