How do you find all the asymptotes for # (3x )/ (x+4) #?

Question

How do you find all the asymptotes for # (3x )/ (x+4)   #?

Sadie Ackerman · Accepted Answer

vertical asymptote x = -4
horizontal asymptote y = 3

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation , let the denominator equal zero. solve : x + 4 = 0 → x = -4 is the asymptote Horizontal asymptotes occur as #lim_(xto+-oo) f(x) → 0# divide terms on numerator/denominator by x → #((3x)/x)/(x/x + 4/x) = 3/(1 + 4/x) # as # x tooo , 4/x to 0# #rArr y = 3/1 = 3 " is the asymptote " # Here is the graph of the function. graph{(3x)/(x+4) [-20, 20, -10, 10]}

Benjamin Achtenberg · Answer

undefined To find the asymptotes of the rational function $ \frac{3x}{x + 4} $, we need to identify any vertical, horizontal, and slant asymptotes.

Vertical Asymptote:
The vertical asymptote occurs when the denominator of the rational function is equal to zero, but the numerator is not. In this case, the vertical asymptote is at $ x = -4 $ because when $ x = -4 $, the denominator becomes zero.

Horizontal Asymptote:
To find the horizontal asymptote, we examine the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, the horizontal asymptote is at the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, the degree of the numerator (1) is less than the degree of the denominator (1), so the horizontal asymptote is at $ y = 0 $.

Slant Asymptote:
A slant asymptote exists when the degree of the numerator is exactly one greater than the degree of the denominator. To find it, perform polynomial long division. Since the degrees of the numerator and denominator are equal, there is no slant asymptote for this function.

In summary, the asymptotes for $ \frac{3x}{x + 4} $ are:

- Vertical asymptote: $ x = -4 $
- Horizontal asymptote: $ y = 0 $
- No slant asymptote.