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Chapter 2 | Limits
Observe that for all values of x (regardless of whether they are approaching a ), the values f ( x ) remain constant at c .We have no choice but to conclude lim x → a c = c . The Existence of a Limit As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist. Example 2.7 Evaluating a Limit That Fails to Exist
sin(1/ x ) using a table of values.
Evaluate lim x →0
Solution Table 2.5 lists values for the function sin(1/ x ) for the given values of x .
⎛ ⎝
⎞ ⎠
⎛ ⎝
⎞ ⎠
1 x
1 x
sin
sin
x
x
−0.1
0.544021110889
0.1
−0.544021110889
−0.01
0.50636564111
0.01
−0.50636564111
−0.001
−0.8268795405312
0.001
0.826879540532
−0.0001
0.305614388888
0.0001
−0.305614388888
−0.00001
−0.035748797987
0.00001
0.035748797987
−0.000001 0.349993504187
0.000001 −0.349993504187
Table 2.5 Table of Functional Values for lim x →0 sin ⎛ ⎝ 1 x
⎞ ⎠
After examining the table of functional values, we can see that the y -values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x -values approaching 0: 2 π , 2 3 π , 2 5 π , 2 7 π , 2 9 π , 2 11 π ,…. The corresponding y -values are 1, −1, 1, −1, 1, −1,…. At this point we can indeed conclude that lim x →0 sin(1/ x ) does not exist. (Mathematicians frequently abbreviate
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