Question

For the following function: g(x) = x^2 + 4x - 5 / x^2 + 7x + 10 (a) Find the domain. Answer .................. (b) horizontal asymptotes? (c) vertical asymptotes ? (d) is there discontinuity ?

Answer 1

`g(x)=(x^2 + 4x - 5) / (x^2 + 7x + 10)=((x-1)(x+5))/((x+2)(x+5))`

Domain
`x!=-2 and x!=-5` as this would make the denominator `0`
As long as `x!=-5` we may cancel out the `(x+5)`'s
We get `g(x)=(x-1)/(x+2)`
There is a discontinuity at `x=-2 and x=-5`
At the `x=-5` discontinuity we may get as close as we want, because value `!=5` would make `(x+5)`'s cancel, and near this point the value of `g(x)` would be near `2`, or as we say:

`lim_(x->-5^+) g(x)=lim_(x->-5^-) g(x)=2` but `g(-5)` is undefined

Vertical asymptotes
`x=-2`, as we can say

`lim_(x->-2^+) g(x)=-oo` and `lim_(x->-2^-) g(x)=+oo`

Horizontal asymptote
As `x` gets larger and larger, the `-1` and `+2` in
`g(x)=(x-1)/(x+2)`matter less and less, so we can say:

`lim_(x->oo) g(x)=lim_(x->-oo) g(x)=1`
graph{(x^2+4x-5)/(x^2+7x+10) [-25.66, 25.66, -12.83, 12.81]}