Math HW
ahmedalmahmoud
formal_write-up_1.pdf
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 function y = f(x) that is defined 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) = −∞
(vii) lim x→−3
f(x) exists
−7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7
x
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
7 f(x)
0
1
2. Consider the following limit.
lim x→0
√ x2 + 9 − 3
x2
(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 √ x2 + 9 − 3
x2 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-defined function.
f(x) =
x2 − 16 x2 −x− 12
if x 6= −3, 4
k if x = 4
where k ∈ R. How should we choose the value of k so that f is 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 = c imply 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.
2
