How do you find asymptotes? - Answers

Veritcal asymptotes are where the denominator of a fraction becomes 0 and the value of f(x) becomes undefined. Set the denominator to 0 and solve.Horizontal asymptotes are values of f(x) when x→∞ and x→-∞Two examples:f(x)=3x² / (x²+1)Vertical:Set (x²+1)=0 and solve for x.x²=-1 has no answers, so there is no vertical asymptote.Horizontal:Divide all terms by the highest power of x to eliminate unimportant valuesdividing by x², you get 3 / (1 + (1/x²))As x→∞ then 1/x² vanishes, leaving 3/1=3, so there is an asymptote at y=3f(x)=(x-3) / (x²+3x)Vertical:Set (x²+3x)=0 and solve. x={0,-3} so there are vertical asymptotes at 0 and -3Horizontal:Divide out by x²(x/x² - 3/x²) / (x²/x² + 3x/x²) as x→∞The terms in the numerator all vanish, making the answer 0, so there is a horizontal asymptote at 0.Enjoy.■A vertical asymptote also exists for the value of x (assuming a function of x) where the function becomes undefined. For instance:f(x) = ln (x). The logarithmic function is not defined for x