Asymptotes Question

Question

Why does the function f(x)= ln(x-1) have a vertical asymptote, but not a horizontal asymptote.

Answer 1

Answer 2

a vertical asymptote is where y is undefined when x=0

y=ln(x-1)

y=ln(1-1) = ln0 is undefined. the vertical asymptote is x=1

same with y=1/x, x=0 is the vertical asymptote

a horizontal asymptote is where y approaches a constant when x approaches infinity or negative infinity

ln(x-1) = c

make both sides the exponent of e

x-1 = e^c

x = e^c +1

c approaches infinity when x approaches infinity. c never approaches a constant as a limit

unlike y=1/x where y approaches 0 as a limit as x approaches infinity or negative infinity