CalcNotes0201-page10

# CalcNotes0201-page10 - lim x ± a f x = L That is a...

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(Section 2.1: An Introduction to Limits) 2.1.9 Existence of Limits A limit exists ± (if and only if, or iff) the limit can be expressed as a single real constant. Otherwise, the limit does not exist (“DNE”) . Later, we will be able to say that a limit is ± (infinity) or ±² (negative infinity) in some cases, but the limit is still nonexistent in those cases. The notation in those cases indicates why the limit does not exist. Two-Sided Limits If a and L are real constants, then lim x ± a fx () = L ± ( lim x ± a ² fx () = L , and
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Unformatted text preview: lim x ± a + f x ( ) = L ). That is, a two-sided limit exists ± the left-hand and right-hand limits exist, and they equal the same real constant. The value of the two-sided limit then equals that constant. If either one-sided limit does not exist (DNE), or if the two one-sided limits exist but are unequal, then the two-sided limit does not exist (DNE). Example 7 (Revisiting Examples 5 and 6) lim x ± 3 ² x + 3 ( ) = 6 , and lim x ± 3 + x + 3 ( ) = 6 , so lim x ± 3 x + 3 ( ) = 6 ....
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