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{XPPMATH 20 "6#%boLimiting~value~that~f(x)~approaches~as~x~appr oaches~2~from~either~side:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&Lim itG6$-%\"fG6#%\"xG/F*\"\"#%*undefinedG/-F(6#F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oLimiting~value~that~f(x)~approaches~as~x~approaches~ 3~from~the~left:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6%-%\"f G6#%\"xG/F*\"\"$%%leftG\"\"\"/-F(6#F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%`oLimiting~value~that~f(x)~approaches~as~x~approaches~3~from~th e~right:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6%-%\"fG6#%\"xG /F*\"\"$%&rightG%*undefinedG/-F(6#F,\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%boLimiting~value~that~f(x)~approaches~as~x~approaches~ 3~from~either~side:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%&LimitG6$-% \"fG6#%\"xG/F*\"\"$%*undefinedG/-F(6#F,\"\"\"" }}{PAGEBK }{PARA 256 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 262 40 "Limits of Piece wise Functions: Example 2" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" 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G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"# %%leftG%&____~G/-F%6%F'F+%&rightGF./-F%6$F'F+F./-F(6#F,%%____G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 &/-%&LimitG6%-%\"fG6#%\"xG/F*\"\"$%%leftG%&____~G/-F%6%F'F+%&rightGF./ -F%6$F'F+F./-F(6#F,%%____G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%RRedefine~the~function~f~so~that~it~i s~continuous:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6(7$%-~~~~~~~~~~~~G2F'! \"#7$F,/F'F.7$F,32F.F'1F'\"\"\"7$F,32F5F'2F'\"\"$7$F,/F'F:7$F,2F:F'" } }{PAGEBK }{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 271 41 "Limits of Piecewise Functions: Exercise 4" }}{PARA 256 "" 0 " " {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 246 "Read through three d ifferent calculus books and find a definition for each one of the foll owing concepts that makes the most sense to you. When you find the one you like the best for each concept, write it in the appropriate place provided below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 20 "Limit Definition 1. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Left~hand~limitG/-%&LimitG6%-%\"fG6#%\"xG/F+%\"aG%%le ftG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 20 "Limit Definition 2. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%1Right~hand~limitG/-%&LimitG6%-%\"fG6#%\"xG/F+%\"aG%&r ightG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 20 "Limit Definition 3. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Two- sided~limitG/-%&LimitG6$-%\"fG6#%\"xG/F+%\"aG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 20 "Limit Definition 4. " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%.The~function~G\"\"\"-%\"fG6#%\"xG F&%5~is~continuous~at~x~GF&%\"aG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Dr. John Pais, Mathematic s Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds. org or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http: //kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 40 "Limits of Piecewise Functions: Example 1 " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1932 "restart :\nwith(plottools):\nn:=1: # Example number\n ntxt:=convert(n,symbol):\n # Define func tion f here below\nf:=unapply(piecewise(0 B'esBf*F1F+7$$!3'Q;v?s%3z$*F1F+7$$!3MnX]ZIQk\"*F1F+7$$!3ki_^7=q]*)F1F+ 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