Question

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

Answer 1

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.

What are the asymptotes of a function f(x)?

• 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

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