Question

Determine the equations of the vertical and horizontal asymptotes, if any, f(x)= x/x-5

Answer 1

Answer:

vertical asymptote. x = 5

horizontal asymptote, y = 1

Step-by-step explanation:

The vertical asymptote of f(x)= x/x-5 is gotten when the denominator x - 5 = 0 ⇒ x = 5.

The horizontal asymptote of f(x)= x/x-5 is gotten when  we find $\lim_{x \to \infty} f(x)$.

So

$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} \frac{x}{x - 5} \\= \lim_{x \to +\infty} \frac{1}{1 - \frac{5}{x} } \\= \frac{1}{1 - \frac{5}{+\infty }} \\= \frac{1}{1 - 0} \\= \frac{1}{1} \\= 1$

$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{x}{x - 5} \\= \lim_{x \to -\infty} \frac{1}{1 - \frac{5}{x} } \\= \frac{1}{1 - \frac{5}{-\infty }} \\= \frac{1}{1 + 0} \\= \frac{1}{1} \\= 1$

Since

$\lim_{n \to +\infty} f(x) = \lim_{n \to -\infty} f(x) = 1. The limit exists\\ horizontal asymptote = \lim_{n \to \infty} f(x) = 1$