Question

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

Answer 1

Using the asymptote concept, it is found that:

• The vertical asymptote is of x = 25.

• The horizontal asymptote is of y = 5.

• Considering the horizontal asymptote, it is found that the end behavior of the function is that it tends to y = 5 to the left and to the right of the graph.

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.

In this problem, the function is:

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

Considering the denominator, the vertical asymptote is:

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 of the function is that it tends to y = 5 to the left and to the right of the graph.

More can be learned about asymptotes and end behavior at https://brainly.com/question/28037814

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