Some stuff about limits

[SuperDude]

Alright, so here's the question:

Given:

g(x) = (a(x+b)(x-c)(x+d))/(e(x-f)(x-b)(x-c)) where a,b,c,d,e, and f are all unique positive real numbers.


At what values of x = h, if any, does the lim as x approaches h of g(x) = ∞ or −∞? Justify your answer.

Alright, so the obvious answer(and the one the teacher was looking for): In order for the limit as defined to exist, there must be a vertical asymptote. This will occur when x = f or x = b. When x = c, you have a factor/cancel limit, or removable discontinuity.

This is what I'm coming up with: However, as x approaches f or b, the limit of the function from the left and right hand side as x approaches these points will be different.

For example, as x approaches f from the left it might be -infinity, but as x approaches f from the right the limit would be +infinity(which way it goes would depend upon whether f is greater than or less than b). Since the left and right hand limits do not equal eachother, the limit as x approaches these points DNE. Therefore, there are no points that satisfy the given limit.

So, what's wrong with my logic? Am I overlooking something or did I do something wrong? TIA, please let me know what you guys get for this.

 

[pka]

What you wrote, “as x approaches f or b, the limit of the function from the left and right hand side as x approaches these points will be different” is true. You should be careful however about what you said about right/left approaches.

 

[SuperDude]

How so? I realize that whether f is greater than or less than b will change whether the left or right hand limits are approaching infinity or -infinity.

Also, what bearing does this all have on the actual question? Was my answer right or was my teacher's answer right? And why?

 

[pka]

I do not understand how you could have misread what I wrote.
I said that you are correct: you statement is true.
I do think that you ought not go into the right/left at b & f.
It should be enough to note that the limit does not exist at either value.