How do you find the vertical, horizontal or slant asymptotes for #(x-5)/(x+1)#?

Answer 1

vertical asymptote x = - 1
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.

the asymptote of x + 1 = 0 is x = -1.

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

Divide each term by x on the numerator and denominator.

# (x-5)/(x+1) = (x/x -5/x)/(x/x + 1/x ) = (1 - 5/x)/(1 + 1/x) #
now as x → ∞ , # 5/x" and " 1/x → 0#
#rArr y = 1 " is the asymptote " #

The function graph{(x-5)/(x+1) [-20, 20, -10, 10]} is shown here.

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Answer 2

To find the vertical asymptote(s), set the denominator equal to zero and solve for x. In this case, the vertical asymptote is x = -1.

To determine if there's a horizontal asymptote, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, then there's a horizontal asymptote at y = 0. Here, the degrees are equal, so there is no horizontal asymptote.

For slant asymptotes, divide the numerator by the denominator using polynomial long division or synthetic division. The quotient obtained represents the slant asymptote. Here, perform polynomial long division to get the slant asymptote, if one exists.

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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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