Find a value of the constant k such that the limit exists

Find a value of the constant k such that the limit exists.

lim (x2 - kx + 4) / (x - 1)

We could do...
just try number until it factors nicely..
k would equal 5.. to give us
x2-5x+4 = (x-1)(x-4)
the (x-1) would cancel .. leaving just x-4.. and the limit would be 1-4 = -3...

Is there an easier way of doing this than just guessing to try to figure out which value of k would make the polynomial factor nicely so that it would cancel with the factor in the denominator? Because, some of the other problems in this section get a little too tough to just be able to spit out the answer...
I'm not sure you could say this is easier. But definately more systematic.

For the problem you just solved, you can break the function as such:


To cancel the bottom, you know one of the roots must be 1 and therefore must have a (x-1) factor on top. Hence let a=-1. Also from the first equation, you know that ab = 4. Therefore b=-4. From the equation, you know that -k=(a+b)=-5, so k=5.
ok, I understand how to do those now.. But, how would I set up something like this?

lim (e2x - 5) / (ekx + 3)
x-> -infinity
Well, try to think about that one logically. What happens to e^2x and e^kx (assuming k is not negative) as x-> -infinity.

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