How do you find vertical, horizontal and oblique asymptotes for #y = (x-4)^2/(x^2-4)#?

Answer 1

vertical asymptotes x = ± 2
horizontal asymptote y = 1

When the denominator of a rational function tends to zero, vertical asymptotes occur. Let the denominator equal zero to find the equation or equations.

solve : # x^2 - 4 = 0 → (x-2)(x+2) = 0#
#rArr x = ± 2 " are the asymptotes " #
Horizontal asymptotes occur as #lim_(xto+-oo) f(x) to 0 #
now numerator = #(x - 4)^2 = x^2 - 8x + 16 #
and y = #(x^2-8x+16)/(x^2-4) #
divide all terms on numerator/denominator by #x^2#
#(x^2/x^2 -(8x)/x^2 + 16/x^2)/(x^2/x^2 - 4/x^2)=(1-8/x+16/x^2)/(1-4/x^2)#
As # xtooo , 8/x , 16/x^2 " and " 4/x^2 to 0#
#rArr y = 1/1 = 1 " is the asymptote " #

There are no oblique asymptotes in this case because oblique asymptotes arise when the degree of the numerator is greater than the degree of the denominator.

This function's graph is shown here: graph{(x-4)^2/(x^2-4) [-10, 10, -5, 5]}

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Answer 2

To find the vertical asymptotes, set the denominator equal to zero and solve for x. To find horizontal asymptotes, analyze the behavior of the function as x approaches positive or negative infinity. For oblique asymptotes, use polynomial long division to divide the numerator by the denominator and examine the quotient.

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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.

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.

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.

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.

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