Question

Can you please help me with 28 please list the end behaviors such as limits and as_,_

Answer 1

N 28

we have the rational function

`f(x)=(x^2-4x+3)/(x^2-4x-5)`

Simplify the numerator

`x^2-4x+3=(x-1)(x-3)`

Simplify the denominator

`x^2-4x-5=(x+1)(x-5)`

substitute in the given function

`f(x)=((x-1)(x-3))/((x+1)(x-5))`

Remember that the denominator cannot be equal to zero

The domain of the rational function is all real numbers except for x=-1 and x=5

we have a vertical asymptote at x=-1 and at x=5

Find out horizontal asymptotes

Degree on Top is Equal to the Bottom

In this case, the graph has a horizontal asymptote along y=1/1=1

at y=1

see the attached figure to better understand the problem

End behavior

as x→−∞ ------> f(x)→1

as x→+∞ ------> f(x)→1

as x→-1 to the left ------> f(x) →+∞

as x→-1 to the right -----> f(x)→−∞

as x→5 to the left -----> f(x) →−∞

as x→5 to the right ----> f(x)→+∞