Limit Examples, Continuity, Horizontal Asymptotes

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Apply a limit to all portions using the squeeze theorem: Useful for limits of the form or with absolute value signs. Theorem: a+ = x -> a and x > a a- = a -> a and x < a. Math 137 page 2 ie the limit dne. We have a vertical asymptote at x=1 b) Ie dne because function gets arbitrarily large as . c) Limits may also not exist due to wild oscillations. Definition: a function f is continuous at x=a if f(a) must be defined at a. If any of these 3 conditions do not hold, then f is discontinuous at x=a. Has a removable discontinuity at is not defined (left and right limits are not equal) So f is discontinuous at x = 1. Theorem: if f and g are continuous at x=a then so is. Proof of 3) f is continuous at a means g is continuous at a means.