High School

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Limits and continuous functions: Recall if a function f is continuous, then lim f(bn) provided lim bn exists.
(a) Find the limit of an = tan-1 3n √9n2+4 if it exists.
(b) Can we use this idea to compute lim 71-00 [cos()] ? Explain why or why not.

Answer :

The limit of an is tan-1(3) as n approaches infinity. For the second part, we need more information to determine the limit of the cosine function as it depends on the angle or function inside the cosine, which is not provided.

For part (a), we need to find the limit of an as n approaches infinity. Limits at infinity can often be solved using l'Hopital's Rule, but in this case, since tan-1 is a continuous function, we can find the limit by simply substituting the expression as n grows large. The expression inside tan-1 approaches 3 as n approaches infinity, so the limit of an is tan-1(3).

For part (b), we cannot use the property that lim f(bn) equals f(lim bn) if f is continuous and the limit of bn exists to compute lim(cos()). This is because cos() is given with no context for what the angle or function inside the cosine is approaching, which is essential for determining its limit. Without additional information, the question of lim 71-00 [cos()] is incomplete and cannot be answered.

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