Chapter 2 MATH180 FALL 2019.pdf - Chapter 2 Limits and...

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Chapter 2: Limits and Derivatives 2.2: The Limit of a Function Example 1: a) Graph the function ࠵? ࠵? = ! ! ! ! ! ! ! and use numerical methods to investigate its behavior near ࠵? = 2 . b) Use the numerical and graphical methods in part (a) to guess the value of lim ! ! ! ! ! ! ! ! ! .
Intuitive Definition of a Limit: Suppose ࠵? ( ࠵? ) is defined when ࠵? is near the number ࠵? . (This means that ࠵? is defined on some open interval that contains ࠵? , except possibly at ࠵? itself.) Then we write lim ! ! ࠵? ࠵? = ࠵? and say “the limit of ࠵? ( ࠵? ) , as ࠵? approaches ࠵? , equals ࠵? if we can make the values of ࠵? ( ࠵? ) arbitrarily close to ࠵? (as close to ࠵? as we like) by restricting ࠵? to be sufficiently close to ࠵? (on either side of ࠵? ) but not equal to ࠵? . Note: lim ! ! ࠵? ࠵? = ࠵? is equivalent to ࠵? ࠵? ࠵? as ࠵? ࠵? , and is read “ ࠵? ( ࠵? ) approaches ࠵? as ࠵? approaches ࠵? ”. Example 2: Estimate the value of each limit graphically . a) lim ! ! !"# ! ! b) lim ! ! sin ! !
Definitions: One-Sided Limits 1. We write lim ! ! ! ࠵? ࠵? = ࠵? and say the left-hand limit of ࠵? ( ࠵? ) as ࠵? approaches ࠵? [or the limit of ࠵? ( ࠵? ) as ࠵? approaches ࠵? from the left ] is equal to ࠵? if we can make the values of ࠵? ( ࠵? ) arbitrarily close to ࠵? by taking ࠵? to be sufficiently close to ࠵? with ࠵? less than ࠵? . Note: lim ! ! ! ࠵? ࠵? = ࠵? is equivalent to ࠵? ࠵? ࠵? as ࠵? ࠵? ! . 2. We write lim ! ! ! ࠵? ࠵? = ࠵? and say the right-hand limit of ࠵? ( ࠵? ) as ࠵? approaches ࠵? [or the limit of ࠵? ( ࠵? ) as ࠵? approaches ࠵? from the right ] is equal to ࠵? if we can make the values of ࠵? ( ࠵? ) arbitrarily close to ࠵? by taking ࠵? to be sufficiently close to ࠵? with ࠵? greater than ࠵? . Note: lim ! ! ! ࠵? ࠵? = ࠵? is equivalent to ࠵? ࠵? ࠵? as ࠵? ࠵? ! . 3. Theorem: lim ! ! ࠵? ࠵? = ࠵? if and only if lim ! ! ! ࠵? ࠵? = ࠵? and lim ! ! ! ࠵? ࠵? = ࠵? . Example 3: Use the graph below to evaluate each limit, or if appropriate indicate that the limit does not exist. a) lim ! ! ! ! ࠵? ࠵? = b) lim ! ! ! ! ࠵? ࠵? = c) lim ! ! ! ࠵? ࠵? = d) lim ! ! ! ! ࠵? ࠵? = e) lim ! ! ! ! ࠵? ࠵? = f) lim ! ! ! ࠵? ࠵? = g) lim ! ! ! ࠵? ࠵? = h) lim ! ! ࠵? ࠵? = i) lim ! ! ! ࠵? ࠵? = j) lim ! ! ! ࠵? ࠵? = k) lim ! ! ࠵? ࠵? = l) lim ! ! ࠵? ࠵? =
Definitions: Intuitive Definition of Infinite Limits Definition: Vertical Asymptote The vertical line ࠵? = ࠵? is called a vertical asymptote of the curve ࠵? = ࠵? ( ࠵? ) if at least one of the following statements is true: lim ! ! ࠵? ࠵? = lim ! ! ! ࠵? ࠵? = lim ! ! ! ࠵? ࠵? = lim ! ! ࠵? ࠵? = lim ! ! ! ࠵? ࠵? = lim ! ! ! ࠵? ࠵? = 1. Let ࠵? be a function defined on both sides of ࠵? , except possibly at ࠵? itself. Then lim ! ! ࠵? ( ࠵? ) = means that the values of ࠵? ( ࠵? ) can be made arbitrarily large (as large as we please) by taking ࠵? sufficiently close to ࠵? , but not equal to ࠵? . Alternatively, lim ! ! ࠵? ( ࠵? ) = can be expressed in the following manner: ࠵? ( ࠵? ) as ࠵? ࠵? . Note: This does not mean that we are regarding as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit does not exist: ࠵? ( ࠵? ) grows without bound as ࠵? approaches ࠵? . 2. Let ࠵? be a function defined on both sides of ࠵? , except possibly at ࠵? itself. Then lim ! ! ࠵? ( ࠵? ) = means that the values of ࠵? ( ࠵? ) can be made arbitrarily large negative by taking ࠵? sufficiently close to ࠵? , but not equal to ࠵? .

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