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Then, no matter what positive epsilon you've chosen, it must be possible to adjust the delta slider so that the graph passes through the yellow region without straying into the green region. We need the smaller of these two distances; The upper bound of 6.25 is 2.25 units from 4.
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We must have δ ≤ 1.75. The second step is using the fact that limx!1 only looks at values of x that aren't 1, for which we can cancel out the factors of (x 1). Note that the explanation is long, but it will take one through all steps necessary to understand the ideas.
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The lower bound of 2.25 is 1.75 units from 4;
An example will help us understand this definition. By direct substitution, the limit is $5$. An example will help us understand this definition. It means that you can set the epsilon slider to any value you like, except zero.
This section introduces the formal definition of a limit. For example, a more correct answer to limx!1(x4. Note that the explanation is long, but it will take one through all. The triangle inequality is used at a key point of the proof, so we first review this key property of absolute value.
Evaluating a limit using the definition.
The rst step is just rewriting the thing whose limit is being taken.
