# Worksheet: Formal Definition of a Limit

In this worksheet, we will practice interpreting the formal epsilon–delta 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 — in terms of — for the largest so that if then .

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