How do you find the asymptotes for #f(x) = (5x)/(x^2-1)#?

Answer 1

vertical asymptotes at x = ± 1
horizontal asymptote at y = 0

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

solve # x^2 - 1 = 0 → (x-1)(x+1) = 0 → x = ± 1 #

Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0 #

If the degree of the numerator is less than the degree of the denominator, as is the case here, numerator degree 1 , denominator degree 2.

Then the equation is y = 0

Here is the graph as an illustration of the asymptotes. graph{5x/(x^2-1) [-10, 10, -5, 5]}

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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.