How do you find all the asymptotes for #R(x)=(3x+5) / (x-6)#?
1 Answer
Horizontal:
Vertical:
Explanation:
The function is defined where the denominator is not zero, which means that all points are ok except for
The possible asymptotes are thus found computing the limits of
-
#lim_{x \to -\infty} R(x) = lim_{x \to -\infty} (x(3+5/x))/(x(1-6/x))#
You can simplify the#x# 's, and both#5/x# and#-6/x# tend to zero. So, the limit is#3# . -
For the limits in
#6# , from left and right, we have that the denominator tends to zero, so the function will tend to positive or negative infinity, depending on the signs: if#x\to 6^{-}# , then the denominator tends to zero from negative values, and the numerator is positive, so the limit is#-infty# . On the contrary, if#x\to 6^{+}# , both numerator and denominator are positive, so the limit is#infty# . -
The limit for
#x\to\infty# requires exactly the same calculations as the one for#x\to-\infty# , so the asymptote will be#y=3# again. -
Since the function has horizontal asymptotes, there can't be oblique ones.