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3-18.
  1. Evaluate each limit. If the limit does not exist, say so but also state if y is approaching positive or negative infinity. Homework Help ✎

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.