(a) Does f (x) have a horizontal asymptote? If so, what is it?(b) Does f (x) have any vertical asymptotes? If so, what are they?

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03-11-2022
Mathematics

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(a) Does f (x) have a horizontal asymptote? If so, what is it?(b) Does f (x) have any vertical asymptotes? If so, what are they?

Sagot :

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HenleyB722795 HenleyB722795 · Answer 1 · 2022-11-03T08:33:37-04:00

a) Horizontal asymptotes are horizontal lines that the graph of a function approaches but never touches. To find the horizontal asymptote, we would apply one of the rules which states that

If the degree of the of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x axis of the graph. It occurs at y = 0

The degree is the largest exponent in the function. Looking at the given function, the degree of the numerator is 2 while the degree of the denominator is 3. Thus,

there is a horizontal asymptote at y = 0

b) The vertical asymptotes are vertical lines which correspond to the zeros of the denominator of rational functions. It is equal to the values of x that make the denominator to be zero. Looking at the given function, (x + 1) cancels out in the numerator and denominator. We are left with (x - 4) and (x + 5). We would equate both terms to zero and solve for x. These values of x would make the denominator to be equal to zero. We have

x - 4 = 0

x = 4

x + 5 = 0

x = - 5

Thus,

there are vertical asymptotes at x = - 5 and x = 4

Sagot :

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