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castortr0y [4]
1 year ago
15

Find the domain, vertical asymptotes, and horizontal asymptotes of the function.

Mathematics
1 answer:
motikmotik1 year ago
3 0

Considering the asymptote concept, we have that:

  • The domain of the function is: (-\infty, -8) \cup (-8, 8) \cup (8, \infty).
  • The vertical asymptotes of the function are of x = -8 and x = 8.
  • The horizontal asymptote is of y = 0.

<h3>What are the asymptotes of a function f(x)?</h3>

  • The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.
  • The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.

For this problem, the function is given by:

f(x) = \frac{x}{x^2 - 64}

The vertical asymptotes are the zeroes of the denominator, hence:

x² - 64 = 0 -> x² = 64 -> x = -8 or x = 8.

Hence the domain is:

(-\infty, -8) \cup (-8, 8) \cup (8, \infty).

The horizontal asymptote is the limit of f(x) as x goes to infinity, hence:

y = \lim_{x \rightarrow \infty} \frac{x}{x^2 - 64} = \lim_{x \rightarrow \infty} \frac{x}{x^2} = \lim_{x \rightarrow \infty} \frac{1}{x} = \frac{1}{\infty} = 0

More can be learned about asymptotes at brainly.com/question/16948935

#SPJ1

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