Can someone help me solve this? Thanks!

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1 Answer
Sep 20, 2017

#"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"#