Chapter 2 | Limits
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Solution Suppose that L is a candidate for a limit. Choose ε =1/2. Let δ >0. Either L ≥0 or L <0. If L ≥0, then let x = − δ /2. Thus, | x −0| = | − δ 2 −0 | = δ 2 < δ and | | − δ 2 | − δ 2 − L | = |−1− L | = L +1≥1> 1 2 = ε . On the other hand, if L <0, then let x = δ /2. Thus, | x −0| = | δ 2 −0 | = δ 2 < δ and | | δ 2 | δ 2 − L | = |1− L | = | L | +1≥1> 1 2 = ε .
Thus, for any value of L , lim x →0 | x |
x ≠ L .
One-Sided and Infinite Limits Just as we first gained an intuitive understanding of limits and then moved on to a more rigorous definition of a limit, we now revisit one-sided limits. To do this, we modify the epsilon-delta definition of a limit to give formal epsilon-delta definitions for limits from the right and left at a point. These definitions only require slight modifications from the definition of the limit. In the definition of the limit from the right, the inequality 0< x − a < δ replaces 0< | x − a | < δ , which ensures that we only consider values of x that are greater than (to the right of) a . Similarly, in the definition of the limit from the left, the inequality − δ < x − a <0 replaces 0< | x − a | < δ , which ensures that we only consider values of x that are less than (to the left of) a .
Definition Limit from the Right: Let f ( x ) be defined over an open interval of the form ( a , b ) where a < b . Then,
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