Very tricky piecewise function problem...

Question

Just looking for a starting point on this question thank you!!! So if the limit as $x$ goes to $-\infty$ is $7$ then we are dealing with the top part of the function???

Answer 1

From $\lim_{x\to-\infty} r(x)=7$, you can deduce a couple of things:

• The degree of the numerator & denominator must be the same; otherwise, the limit would be $0$ if the denominator had greater degree / $\infty$ if the numerator had greater degree
• You already have a degree of $4$ in the denominator, so that means to get a degree of $4$ in the numerator, you have to have $A = 2$.
• In the limit as $x\to-\infty$ you will have to end up with something like $7x^4/x^4$ if you want to get a limit of $7$. The constants and the terms of lesser degree will vanish in the limit. This gives you a hint about what the coefficient on the $x^4$ term will have to be in the numerator, in order to get $7$.

For the other parts of the problem - you can only get a vertical asymptote at some $x=k$ if you have a term like $(x-k)$ in the denominator (or something else where plugging in $x=k$ will cause the whole denominator to be $0$), that does not get cancelled out in the numerator - because, if it gets cancelled, you will just have a removable singularity.

And to make the whole function continuous, you will need $$ \lim_{x\to 4^{+}} r(x) = \lim_{x\to 4^{-}} r(x) $$ i.e. the limit of the first piece, approached from the left, has to be the same as the limit of the second piece, approached from the right.