**Unformatted text preview: **09/22/2000 FRI 11:16 FAX 6434330 MOFFITT LIBRARY .001 Math HlA Fall, 1995
Professor K. A. Ribet
Midterm Exam—September 28, 1995 This is an “open book” exam. You may consult your notes and textbook. Grading is
based on completeness, clarity, and accuracy. Please write in complete English sentences
and explain your reasoning carefully. 1 (4 points). Find the domain of 9(x) = \/.’L‘4 "-— 3132 — 4. 2 (6 points). Use mathematical induction to prove that +L+_1_+...+m_ n
2-3 3.4 n(n+1)“n+1 L
1-2
foralln21. 3 (5 points) Show that the axioms P10—12 on page 9 on Spivak imply that a2 is positive
for all non-zero real numbers a. [The ﬁrst axiom states, for each real number a, that
exactly one of the following holds: (i) a : 0; (ii) a > 0; (iii) a < 0. The other two state
that the sum or product of two positive numbers is positive] Using the axioms, prove that
—l is not the square of a real number. 4 ( 8 points). Give a detailed proof from ﬁrst principles (i.e., starting from the deﬁnition
of “‘limit”) that m2 —> 9 as a: —> 3. 5 (5 points). For each real number It, deﬁne ft (:0) by the recipe ft($)={—m+2 if$>t 353—4 ifccﬁt. For which 15 does limt ft(sc) exist? Explain your reasoning without giving a formal proof.
12—} 6 {’7 points). Exhibit a function f(x) deﬁned on all of R with the following property: For
each M > 0 there is a 5 > 0 such that ﬁns) 2 M for all :5 with 0 < |az| < 6. Show that this property implies that 1i_r+r(1)f(m) does not exist.
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