Question

Can someone help me solve this? Thanks!

Answer 1

`"point discontinuity at "(5,1/4)" removable"`
`"vertical asymptote at "x=1" non-removable"`

Explanation:

`"factorise and simplify"`

`y=(cancel((x-5)))/(cancel((x-5))(x-1))=1/(x-1)`

`"since we have removed the factor "(x-5)`
`"from the numerator/denominator this indicates a"`
`"hole"`

`x-5=0rArrx=5" and "y=1/(5-1)=1/4`

`rArr"point discontinuity at "(5,1/4)larr" removable"`

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

`"solve "x-1=0rArrx=1" is the asymptote"`

`" the asymptote is non-removable"`
graph{1/(x-1) [-10, 10, -5, 5]}

`"note the graph of " y=1/(x-1)" is the same as "`

`"the graph of "y=(x-5)/(x^2-6x+5)" but without the hole"`