Let an→0, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined): limn→∞1+2an1+3an−4(an)2

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Let an→0, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined):  limn→∞1+2an1+3an−4(an)2

opsadnojD · Accepted Answer

To compute, limn→∞1+2an1+3an−4(an)2  Given that, limn→∞an=0  Algebraic limit theorem: Let an and bn be sequences of real numbers such that  limn→∞an=a and limn→∞bn=b, Then following statements hold.  a) limn→∞c⋅an=c⋅a  b) limn→∞(an+bn)=a+b  c) limn→∞(an⋅bn)=a⋅b  d) If bn≠0, for all n, then limn→∞anbn=ab  Using the algebraic limit theorem,  limn→∞1+2an(1+3⋅an−4(an)2  =limn→∞1+limn→∞2anlimn→∞1+limn−&amp;gt;∞3an−limn→∞4(an2  =limn→∞1+2⋅limn→∞anlimn→∞1+limn→∞an−4limn→∞(an)2  Using given limn→∞an=0  limn→∞1+2an1+3an−4(an)2  =1+2⋅(0)1+3⋅(0)−4⋅(0)2  =11  =1  Therefore, limn→∞1+2an1+3an−4(an)2=1

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}

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