Worksheet: Formal Definition of a Limit
In this worksheet, we will practice interpreting the formal epsilonβdelta definition of a limit and finding the value of delta for a given epsilon.
Q1:
Find the largest such that if , then . Give your answer as a fraction.
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Q2:
Find the largest such that if , then . Give your answer as a fraction involving .
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Q3:
Find the largest such that if , then . Give your answer as a fraction involving and .
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Q4:
In the figure, we have the graph that is increasing and concave up. Next to are the points and for a small .
Which point is closer to along the horizontal axis, or ?
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What is the distance between the nearest point and from your answer above? Give an expression that involves and absolute values.
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Use your answers above to find such that whenever . Give your answer to 4 decimal places.
Q5:
The figure shows the graph of around a point where and . Nearby are points and .
From the graph, looking at the -axis, which of the points and is nearest to ? Let this number be .
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What is in terms of ?
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The claim is that if and , then provided that is true for all small . What is the largest such ?
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The condition was to ensure truth to the figure. Does the same work if ?
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It would appear that all your arguments depended upon the fact that the graph of the function was concave up. Let . Find so that whenever . Assume that is positive.
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Q6:
Sometimes, we do not have concavity to help determine an explicit that shows continuity at a point. The figure shows the graph of
together with its tangent line at the inflection point .
What is for ?
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What is for ?
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By considering the graph near , find the largest so that whenever .
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Q7:
The graph of is concave down and decreasing when . We want to find the maximal so that, for a given , it follows that if then . This will, of course, be a function of .
What is the inverse function near ? Give an expression for .
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Using the concavity of the graph of , determine which point is closer to 3, or , along the horizontal axis. (Do not evaluate them and consider only small .)
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From your answers above, find an expression β in terms of β for the largest so that if then .
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