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Hello!
1. Let us start from the first question, as it is the most straightforward.


(a) As we see, when x approaches positive infinity, f(x) approaches -2.


(b) As x approaches negative infinity, f(x) approaches 2.


(d) As x approaches 3 from any side, f(x) approaches negative infinity, -\infty.
(e) The vertical asymptotes can be found at x=1 and x=3 (finite values of x where f tends to infinity). The horizontal asymptotes can be found at y=2 and y=-2 (the limits of f on infinity, if it exists exist).


2. To determine the limit, multiply and divide the given expression by sqrt(x^2+cx)+sqrt(x^2+dx). The numerator will become (x^2+cx)-(x^2+dx)=x(c-d).
Now divide both numerator and denominator by x and obtain the following:
(c-d)/(sqrt(1+c/|x|)+sqrt(1+d/|x|)).
Now, we are able to let x-gt\infty, and we get the following:
(c-d)/(sqrt(1+0)+sqrt(1+0))=(c-d)/2.
This is the answer.
3. To get the vertical asymptote x=4, put (x-4) into the denominator, as it will give infinity at x=4. To get the vertical asymptote x=6, multiply the denominator by (x-6).
To balance these factors and get the horizontal asymptote y=4, we need the limit of f at infinity to be equal to 4. Place 4x^2 into the numerator:
f(x)=(4x^2)/((x-4)(x-6)).
It is an answer (there are many others possible).