How do you find the vertical, horizontal or slant asymptotes for #f(x) = (x-3)/(x-2)#?

Answer 1

vertical asymptote x = 2
horizontal asymptote y = 1

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : x - 2 = 0 → x = 2 is the asymptote

Horizontal asymptotes occur as

#lim_(xto+-oo),f(x)toc" (a constant)"#

divide terms on numerator/denominator by x

#(x/x-3/x)/(x/x-2/x)=(1-3/x)/(1-2/x)#

as #xto+-oo,f(x)to(1-0)/(1-0)#

#rArry=1" is the asymptote"#

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( both of degree 1 ) Hence there are no slant asymptotes. graph{(x-3)/(x-2) [-10, 10, -5, 5]}

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Answer 2

Benjamin Achtenberg

To find the vertical asymptote of ( f(x) = \frac{x-3}{x-2} ), you set the denominator equal to zero and solve for ( x ). Here, the vertical asymptote occurs where the denominator is zero: ( x - 2 = 0 ), so ( x = 2 ).

There are no horizontal or slant asymptotes for this function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.