How do you find the vertical, horizontal or slant asymptotes for #g(x)=(x+3) /( x(x-5))#?

Question

Ariana Alvarez · Accepted Answer

vertical asymptotes x = 0 , x = 5
horizontal asymptote y = 0

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero. solve : x(x-5) = 0 → x = 0 , x=5 are the equations. Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0# If the degree of the numerator is less than the degree of the denominator, as is the case here, numerator degree 1 and denominator degree 2, then the equation is y = 0. #rArr y = 0 " is the equation "# Here is the graph of the function. graph{(x+3)/(x(x-5)) [-10, 10, -5, 5]}

Christian Antunez · Answer

undefined To find the vertical asymptotes of the function $g(x) = \frac{x + 3}{x(x - 5)}$, we identify the values of $x$ where the denominator is equal to zero but the numerator is not zero.

To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$. If the degree of the numerator is equal to the degree of the denominator, we divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find any slant asymptotes, if they exist, we perform polynomial long division or synthetic division to divide the numerator by the denominator. If the result is a polynomial function plus a proper rational function, the slant asymptote is the polynomial function.

After performing these steps, the vertical asymptotes for $g(x)$ are $x = 0$ and $x = 5$, the horizontal asymptote is $y = 0$, and there are no slant asymptotes.