Section+3.1+-+Introduction+to+Limits

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1 SECTION 3.1 - INTRODUCTION TO LIMITS I. INTRODUCTION TO LIMITS Limits describe the behavior of functions ࠵? = ࠵?(࠵?) near a specified ࠵? -value. Example 1: Consider the function ࠵?(࠵?) = ! ! "# !"# . a. What happens to ࠵?(࠵?) when ࠵? = 1 ? b. What happens to ࠵?(࠵?) when ࠵? c. What happens to ࠵?(࠵?) when ࠵? is close to 1, but less than 1? is close to 1, but greater than 1? ࠵? ࠵?(࠵?) 0 1 0.9 1.9 0.99 1.99 0.999 1.999 ࠵? ࠵?(࠵?) 2 3 1.1 2.1 1.01 2.01 1.001 2.001
2 Important Notes: 1. If no side indicated, the "limit" is implied to be the (two-sided) limit. 2. If lim !→% ࠵?(࠵?) = ࠵? and ࠵? is a finite number , we say ________________________. Otherwise, ______________________________. 3. Limits describe the behavior of ࠵?(࠵?) near ࠵? = ࠵? , not at ࠵? = ࠵? . In fact, the function value ࠵?(࠵?) need not exist for the limit to exist. ONE-SIDED AND (TWO-SIDED) LIMITS 1. If ࠵?(࠵?) becomes arbitrarily close to ࠵? whenever ࠵? gets closer to (but to the left of) ࠵? on the number line, then ࠵? is the left-hand limit . We write lim !→% " ࠵?(࠵?) = ࠵? 2. If ࠵?(࠵?) becomes arbitrarily close to ࠵? whenever ࠵? gets closer to (but to the right of) ࠵? on the number line, then ࠵? is the right-hand limit . We write lim !→% # ࠵?(࠵?) = ࠵? 3. If ࠵?(࠵?) becomes arbitrarily close to a single number ࠵? as ࠵? approaches ࠵? from either side , then the limit of ࠵?(࠵?) as ࠵? approaches ࠵? is ࠵? . We write lim !→% ࠵?(࠵?) = ࠵? or "࠵?(࠵?) → ࠵? as ࠵? → ࠵?" THEOREM The limit itself exists if and only if both one-sided limits exist and are equal. Otherwise, the limit does not exist. Þ In other words, in order for a limit to exist ,
3 Example 2: Find each limit using the graph of ࠵?(࠵?) shown below. a. lim !→"& " ࠵?(࠵?) = e. lim !→"# " ࠵?(࠵?) = i. lim !→# " ࠵?(࠵?) = b. lim !→"& # ࠵?(࠵?) = f. lim !→"# # ࠵?(࠵?) = j. lim !→# # ࠵?(࠵?) = c. lim !→"& ࠵?(࠵?) = g. lim !→"# ࠵?(࠵?) = k. lim !→# ࠵?(࠵?) = d. ࠵?(−2) = h. ࠵?(−1) = j. ࠵?(1) = II. INFINITE LIMITS Example 3: Consider the function ࠵?(࠵?) = # !"& whose graph is shown below. a. What is ࠵?(2) ? b. What happens to ࠵?(࠵?) as ࠵? → 2 " ? c. What happens to ࠵?(࠵?) as ࠵? → 2 ' ? d. What is lim !→& ࠵?(࠵?) ?
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