How do you find the vertical, horizontal or slant asymptotes for #y=(13*x)/(x+34)#?

Question

Anna Allen · Accepted Answer

#"vertical asymptote at "x=-34#
#"horizontal asymptote at "y=13# The denominator cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote. #"solve "x+34=0rArrx=-34" is the asymptote"# #"horizontal asymptotes occur as "# #lim_(xto+-oo),ytoc"( a constant)"# #"divide terms on numerator/denominator by x"# #y=((13x)/x)/(x/x+(34)/x)=13/(1+34/x)# as #xto+-oo,yto13/(1+0)# #rArry=13" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes. graph{(13x)/(x+34) [-80, 80, -40, 40]}

Sadie Ackerman · Answer

undefined To find the vertical asymptote, set the denominator equal to zero and solve for $x$. In this case, the vertical asymptote occurs at $x = -34$.

For horizontal or slant asymptotes, we compare the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (1), we have a slant asymptote. To find it, perform polynomial long division. The quotient represents the equation of the slant asymptote.

Dividing $13x$ by $x + 34$, we get:

$$
\begin{array}{c|c}
13 & x \
\hline
x + 34 & \downarrow \
 & 13x + 442 \
\hline
 & - 442 \
\end{array}
$$

So, the slant asymptote is $y = 13$.