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

Answer 1

Vertical: #uarr x = 2 darr #. Seldom realized oblique asymptote:
y = x+ 6. These are the asymptotes of the hyperbola represented by the equation. See illustrative graph.

Rearrange and cross-multiply to create the form

#x(y-x)-2y-4x+2=0#. This suggest the form
#(x+a)(y-x+b=c# and we have
#(x-2)(x-y+6)=-14# that represents the hyperbola with asymptotes
#x-2 = 0 and x-y+6=0#.

Now search the graph for the asymptotes.

graph{y(x-2)-x^2-4x+2=0 [-39.6, 39.6]} -79.2, 79.2

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Answer 2

To find the vertical asymptote of ( \frac{x^2 + 4x - 2}{x - 2} ), identify any values of ( x ) that make the denominator equal to zero. In this case, ( x - 2 = 0 ) when ( x = 2 ). Therefore, there is a vertical asymptote at ( x = 2 ).

To find the horizontal asymptote, determine the degrees of the numerator and the denominator. Since the degree of the numerator (2) is equal to the degree of the denominator (1), divide the leading coefficient of the numerator by the leading coefficient of the denominator. Thus, the horizontal asymptote is ( y = x^2/x = x ).

For oblique asymptotes, divide the numerator by the denominator using long division or synthetic division. The result will be the equation of the oblique asymptote.

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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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