Let a_nrightarrow0, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined): lim_{nrightarrowinfty}frac{1+2a_n}{1+3a_n-4(a_n)^2}

Annette Arroyo 2020-10-28 Answered
Let an0, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined):
limn1+2an1+3an4(an)2
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Expert Answer

opsadnojD
Answered 2020-10-29 Author has 95 answers
To compute, limn1+2an1+3an4(an)2
Given that, limnan=0
Algebraic limit theorem: Let an and bn be sequences of real numbers such that
limnan=a and limnbn=b, Then following statements hold.
a) limncan=ca
b) limn(an+bn)=a+b
c) limn(anbn)=ab
d) If bn0, for all n, then limnanbn=ab
Using the algebraic limit theorem,
limn1+2an(1+3an4(an)2
=limn1+limn2anlimn1+limn>3anlimn4(an2
=limn1+2limnanlimn1+limnan4limn(an)2
Using given limnan=0
limn1+2an1+3an4(an)2
=1+2(0)1+3(0)4(0)2
=11
=1
Therefore, limn1+2an1+3an4(an)2=1
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