Question

Show all work to identify the asymptotes and end behavior of the function f(x)= 5x/x-25.

Answer 1

Considering the given function and the asymptote concept, we have that:

• The vertical asymptote is of x = 25.

• The horizontal asymptote is of y = 5.

• The end behavior is that as $x \rightarrow \infty, f(x) \rightarrow 5$.

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. Hence it also gives the end behavior of the function.

In this problem, the function is:

$f(x) = \frac{5x}{x - 25}$.

The vertical asymptote is found as follows:

x - 25 = 0 -> x = 25.

The horizontal asymptote is found as follows:

$y = \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{5x}{x - 25} = \lim_{x \rightarrow \infty} \frac{5x}{x} = \lim_{x \rightarrow \infty} 5 = 5$.

Hence the end behavior is that as $x \rightarrow \infty, f(x) \rightarrow 5$.

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

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