4:28 A.9_Written_MATH 1000_2023W TOPIC 1-3: Continuity Q.#1- Q.#6 Please use the graph of function f(x) shown at right to complete the table found below. Final Answers Q.#1 am-1 a=0 a=3 a=5 a=8 a=10 31% -5 -3 1 3 7 9 11 Q.#2 Q.#3 Q.#4 Q.#5 Lim f(x) Lim f(x) Lim f(x) Continuous 2 Q.#6 Differentiable? DX +DA-X f(a) X Q.#5(a) Yes/No? Q.#5(b) Why? Q.#6(a) Yes/No? Q.#6(b) Why? Page 1 of 8 Final Answer (x+1 if x 2 constant "b" that makes g(x) continuous at x=2 Q. #8-Q.# 10 Please use the function shown at the right for these questions. Q.#8 For this function, please do the following. a) Use the definition of continuity to determine algebraically if f(x) is continuous at x 1. for x -1 x -4 Final Answer (yes or no) b) If a discontinuity at x=-1, classify it as removable or non-removable. If non-removable, is it jump or infinite? Final Answer (circle) No discontinuity Removable 4:29 A.9_Written_MATH 1000_2023W 031% b) If a discontinuity at x=1, classify it as removable or non-removable. If non-removable, is it jump or infinite? Final Answer (circle) No discontinuity Removable Non-removable: Jump, Inf. Page 2 of 8 Q.#9 For above function, please find and classify any other removable or non-removable discontinuities elsewhere in the function. If non-removable, classify them as jump or infinite. Hint: Normally one checks transition point(s) and the functions themselves. Q. #8 checked the transition point (x-1), so leaves only the functions themselves. Final Answers Function Continuous where applies? (Y/N) Discontinuous where applies: At x= Removable Non-removable: Jump Non-removable: Inf. a) f(x)= 3 1-x for x-1 b) f(x) = 1 for x=-1 x-2 c) x-4 for x>-1 Q.# 10 Using any discontinuities in Q. #8 - Q# 9, please list the intervals of continuity for f(x). Example: If a function has removable discontinuity at x=2, its intervals of continuity are (-, 2) U(2,+00) or, equivalently, {x|x 2} or {x|x #2} TOPIC 4-5: L'Hpital's Rule Q. # 11 Please evaluate Lim x-25 x+5-4x-x a) Using L'Hpital's Rule Final Answer Page 3 of 8 as follows. Final Answer 4:29 A.9_Written_MATH 1000_2023W 031% c) Lim xex 8818 d) Lim (x-1) tan 3+1+ e) Lim (cosx)(1/x2) +0+x f) Lim (secx- tan x) XT/2 Hint: secx & tanx can be expressed in sinx &/or cosx. Final Answer Final Answer Page 6 of 8 Final Answer Final Answer 4:29 A.9_Written_MATH 1000_2023W 031% x-25 Q. # 11 Please evaluate Lim as follows. x+ 5-4x-x a) Using L'Hpital's Rule b) Using earlier limit technique without L'Hpital's Rule Final Answer Final Answer Q. # 12 Please evaluate Lim- tan(7x) 0 sin(4x) as follows, Final Answer a) Using L'Hpital's Rule Final Answer b) Using earlier limit technique without L'Hpital's Rule Page 4 of 8 Q. #13 Please evaluate the following limits, using L'Hpital's Rule if it applies. If not, please evaluate the limit in some other way. a) Lim x0+ sin x-x Final Answer b) Lim x+0x Final Answer