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Exercise3.1.9: Letc1 beaclusterpointofA⊂Randc2 beaclusterpointofB⊂R. Suppose f: A→B and g: B → R are functions such that f(x) → c2 as x → c1 and g(y) → L as y → c2. If c2 ∈ B also suppose that g(c2) = L. Let h(x) := g

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iven that A and 8 nespectively and cduster amd C 2 anl peiuty f: A B such that B-R Lim Lim gy) L and C 2 avd てつーR We hawe to Lim hix) L show that where luter know that is As we C2 point of must aset B, then thsee Condittioms hold for 8R (i) amust i) the imit of at c2 must exist in R (ii) there tuoo values mut Continuous at C to be be datined at c equal be a is continous ot c Lim gc) Hemce Aince .7= (7)B Lim itix)) dm f also know that We provided that a is continuous at KE Cinfixt しim hex)= So /Lim Li gce)= Lim h tx)L Hemee

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Exercise3.1.9: Letc1 beaclusterpointofA⊂Randc2 beaclusterpointofB⊂R. Suppose f: A→B and g: B → R are functions such that...

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Answer 1

iven that A and 8 nespectively and cduster amd C 2 anl peiuty f: A B such that B-R Lim Lim gy) L and C 2 avd てつーR We hawe to Lim hix) L show that where luter know that is As we C2 point of must aset B, then thsee Condittioms hold for 8R (i) amust i) the imit of at c2 must exist in R (ii) there tuoo values mut Continuous at C to be be datined at c equal be a is continous ot c Lim gc) Hemce Aince .7= (7)B Lim itix)) dm f also know that We provided that a is continuous at KE Cinfixt しim hex)= So /Lim Li gce)= Lim h tx)L Hemee