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Chapter 2 | Limits
Choose δ =min ⎧ ⎩ ⎨ δ 1 , δ 2 ⎫ ⎭ ⎬ . Assume 0< | x − a | < δ . Thus,
0< | x − a | < δ 1 and0< | x − a | < δ 2 .
Hence,
| ⎛ ⎝ f ( x )+ g ( x ) ⎞
⎠ −( L + M ) | = | ⎛
⎝ f ( x )− L ⎞ ⎛ ⎝ g ( x )− M ⎞ ⎠ | ≤ | f ( x )− L | + | g ( x )− M | < ε 2 + ε 2 = ε . ⎠ +
□ We now explore what it means for a limit not to exist. The limit lim x → a
f ( x ) does not exist if there is no real number L for
f ( x ) = L . Thus, for all real numbers L , lim x → a
f ( x ) ≠ L . To understand what this means, we look at each part
which lim x → a
f ( x ) = L together with its opposite. A translation of the definition is given in Table 2.10 .
of the definition of lim x → a
Definition
Opposite
1. For every ε >0,
1. There exists ε >0 so that
2. there exists a δ >0, so that
2. for every δ >0,
3. if 0< | x − a | < δ , then | f ( x )− L | < ε .
3. There is an x satisfying 0< | x − a | < δ so that | f ( x )− L | ≥ ε .
Table 2.10 Translation of the Definition of lim x → a
f ( x ) = L and its Opposite
Finally, we may state what it means for a limit not to exist. The limit lim x → a f ( x ) does not exist if for every real number L , there exists a real number ε >0 so that for all δ >0, there is an x satisfying 0< | x − a | < δ , so that | f ( x )− L | ≥ ε . Let’s apply this in Example 2.43 to show that a limit does not exist. Example 2.43 Showing That a Limit Does Not Exist
| x | x does not exist. The graph of f ( x ) = | x |/ x is shown here:
Show that lim x →0
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