(1) Evaluate the following limits. State the asymptotes, if any:
(i)
(ii)

Let’s evaluate the limits step by step, starting with the first one.

(i) To find the limit , we first substitute into the denominator:

This gives us a in the denominator. We then need to factor the denominator. The polynomial can be factored as follows:

Thus, the limit can be rewritten as:

We can now cancel in the numerator and denominator (noting that we are approaching , so we are not actually at 3):

Substituting :

Thus, the limit is and we note that there is a vertical asymptote at since the denominator equals zero there.

(ii) Now we evaluate .

To simplify for large negative , we factor out from the square roots in both the numerator and denominator.

In the numerator:

Thus, we rewrite the numerator:

In the denominator:

Thus, the denominator becomes:

Now we can simplify the limit:

[
\lim_{x \rightarrow -\infty} \frac{x(1 + \sqrt{1 - \frac{1}{x} + \frac{1}{x^2}})}{2x(1 + \sqrt{1 + \frac{5}{4x} - \frac{1}{4x^2}})} = \lim_{x \rightarrow -\infty} \frac{1 + \sqrt{1 - \frac{1}{x} + \frac{1}{x^2}}}{2(1 + \sqrt{1 + \frac{5}{4x} - \