What determines a horizontal asymptote?

Answer 1

Ezekiel Alexander

Definition

The line #y=b# is called a horizontal asymptote of #f# if at least one of the following limits holds:

#lim_{x to -infty}f(x)=b# or #lim_{x to infty}f(x)=b#.

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Answer 2

Adam Adkins

The horizontal asymptote of a function is determined by the behavior of the function as the input approaches positive or negative infinity. It can be determined by analyzing the degrees of the numerator and denominator polynomials in a rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal 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.