Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Please verify your account before proceeding.

In this lesson, we will learn how to utilize the formal definition of a limit to find the value of delta for a given epsilon.

Q1:

Sometimes, we do not have concavity to help determine an explicit πΏ that shows continuity at a point. The figure shows the graph of

together with its tangent line π¦ = π₯ + 3 at the inflection point ( 0 , 3 ) .

What is π ( π₯ ) β 1 for π₯ > 0 ?

What is π ( π₯ ) β 1 for π₯ β€ 0 ?

By considering the graph near ( 0 , 3 ) , find the largest πΏ so that | π ( π₯ ) β 3 | < π whenever | π₯ | < πΏ .

Q2:

The graph of π ( π₯ ) = 5 β ( π₯ β 2 ) 2 is concave down and decreasing when π₯ > 2 . We want to find the maximal πΏ so that, for a given π > 0 , it follows that if | π₯ β 3 | < πΏ then | π ( π₯ ) β π ( 3 ) | < π . This πΏ will, of course, be a function of π .

What is the inverse function near π₯ = 3 ? Give an expression for π ( π₯ ) β 1 .

Using the concavity of the graph of π , determine which point is closer to 3, π ( 4 + π ) β 1 or π ( 4 β π ) β 1 , along the horizontal axis. (Do not evaluate them and consider only small π .)

From your answers above, find an expression β in terms of π β for the largest πΏ so that if | π₯ β 3 | < πΏ then 4 β π < π ( π₯ ) < 4 + π .

Q3:

Find the largest π > 0 such that if | π₯ β π | < π , then | | | 1 π₯ β 1 π | | | < π . Give your answer as a fraction involving π and π .

Q4:

In the figure, we have the graph π¦ = π ( π₯ ) that is increasing and concave up. Next to ( π₯ , π¦ ) are the points ( π , π¦ + π ) and ( π , π¦ β π ) for a small π > 0 .

Which point is closer to π₯ along the horizontal axis, π or π ?

What is the distance πΏ between the nearest point and π₯ from your answer above? Give an expression that involves π and absolute values.

Use your answers above to find πΏ such that | π β 1 | < 0 . 1 π₯ whenever | π₯ β 0 | < πΏ . Give your answer to 4 decimal places.

Q5:

The figure shows the graph of π ( π₯ ) = 2 π₯ around a point ( π₯ , π¦ ) where π₯ > 0 and π¦ = π ( π₯ ) . Nearby are points οΌ π , π¦ + 1 2 ο and οΌ π , π¦ β 1 2 ο .

From the graph, looking at the π₯ -axis, which of the points π and π is nearest to π₯ ? Let this number be π .

What is π in terms of π₯ ?

The claim is that if π₯ > 0 and 2 π₯ > 1 2 , then | | | 2 π₯ β 2 π₯ | | | < 1 2 β provided that | π₯ β π₯ | < πΏ β is true for all small πΏ . What is the largest such πΏ ?

The condition 2 π₯ > 1 2 was to ensure truth to the figure. Does the same πΏ work if π₯ β₯ 4 ?

It would appear that all your arguments depended upon the fact that the graph of the function π was concave up. Let π ( π₯ ) = π β π₯ . Find πΏ > 0 so that | π ( π₯ ) β π ( π₯ ) | < 1 4 β whenever | π₯ β π₯ | < πΏ β . Assume that π₯ is positive.

Q6:

Find the largest πΏ > 0 such that if | π₯ β 5 | < πΏ , then | | | 1 π₯ β 1 5 | | | < 1 1 0 . Give your answer as a fraction.

Q7:

Find the largest πΏ > 0 such that if | π₯ β 5 | < πΏ , then | | | 1 π₯ β 1 5 | | | < π . Give your answer as a fraction involving π .

Donβt have an account? Sign Up