Factor the numerator and denominator. If you can cancel out an (x−2), then the limit exists. If not, then it DNE.
You could factor the denominator. Or you could multiply the top and bottom by the conjugate of
The (x−2) in the denominator will not cancel out. This means that there is a vertical asymptote at x = 2. How can we determine if the asymptote approaches +∞ or −∞ as x → 2− from the left?
Choose a value that is to left of x = 2, and evaluate. Note: we only need to determine if each term is positive or negative. Choose x → 1.9:
DNE but y → −∞.
This is a limit x→∞. That means we are looking at end behavior. Is there a horizontal asymptote, or do the y-values approach ∞ or −∞?
Consider only the terms with the highest power in the numerator and the denominator. Also consider their coefficients, and their signs.