Math HW

MTH 251 Name:

Summer 2017 Formal Write-up #1 (out of 20 pts) { Due Monday, June 10 Please do the following problems on a separate sheet of paper. A few notes:

I will be checking for organization, conceptual understanding, and proper mathematical communica- tion, as well as completion of the problems.

Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.

Use correct mathematical notation. Include \=" and \ " where appropriate.

You may work with your classmates. However, please submit your own work!

Please use this sheet as a cover page.

1.Graph a functiony= f(x ) that is de ned on all numbers 7 x 7 and has the following properties.

You may do this on this sheet! [7 points] (i)lim x! 0+ f (x ) = 4 (ii)lim x! 3 f (x ) = 1 (iii)lim x! 0 f (x ) = 2 (iv) f(0) = 4 (v) f( 3) = 0 (vi)lim x! 2+ f (x ) = 1 (vii)lim x! 3f (x ) exists 7 6 5 4 3 2 1 1 2 3 4 5 6 7 x 7 6 5 4 3 2 1 1234567 f (x ) 0 1 2.Consider the following limit.

lim x ! 0p x 2 + 9 3 x 2 (a)Investigate the above limit numerically. For this, complete the following table. You may do this on this sheet! x 0:0001 0:00005 0:00001 0 0.00001 0.00005 0.0001 p x 2 + 9 3 x 2 undef.

Based on your numerical investigation, estimate the value of the given limit. [3 points] (b)Investigate the above limit algebraically. (Hint : Multiply both the numerator and denominator of the given function by the conjugate of the numerator.) [3 points] (c)Do the results in (a) and (b) agree? If not, which investigation (numerical or algebraic) is more trustworthy? Comment on it. [2 points] 3.Consider the following piecewise-de ned function.

f(x ) = 8 > < > : x 2 16 x 2 x 12 if x6 = 3; 4 k ifx= 4 where k2 R. How should we choose the value of kso that fis continuous at x= 4? [5 points] EC Problem (Up to 2 points) Problem. Discuss how the continuity (at a point, say x= c) and the existence of the limit of a function (as x! c) relate. Does one imply the other? That is, does the continuity of a function at x= cimply the existence of the limit as x! c? What about the converse? Which condition is stronger/weaker? For the weaker implication, provide a counterexample to justify your claim.