Suppose that limn →∞ an=1, limn →∞ bn=-1, and 0 < -bn < an for all n . Evaluate each of the

step 1 We're given the limit of the AN -nguess to infinity equal to 1. Also the limit of the PN equal t

Suppose that limn →∞ an=1, limn →∞ bn=-1, and 0

Suppose that $\lim _{n \rightarrow \infty} a_{n}=1, \quad \lim _{n \rightarrow \infty} b_{n}=-1, \quad$ and $0 < -b_{n} < a_{n}$ for all $n .$ Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. $$\lim _{n \rightarrow \infty} \frac{a_{n}+b_{n}}{a_{n}-b_{n}}$$

Suppose that $\lim _{n \rightarrow \infty} a_{n}=1, \quad \lim _{n \rightarrow \infty} b_{n}=-1, \quad$ and $0 < -b_{n} < a_{n}$ for all $n .$ Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists.

$$\lim _{n \rightarrow \infty} \frac{a_{n}+b_{n}}{a_{n}-b_{n}}$$

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Key Concepts

Limit Arithmetic

Limit arithmetic involves the operations of addition, subtraction, multiplication, and division performed on sequences. It states that if individual sequences converge, then the sum, difference, or product of those sequences also converges to the sum, difference, or product of their limits. Care must be taken with division, where the limit of the denominator must not be zero.

Quotient of Limits

The quotient of limits is a specific principle within limit arithmetic where, if two sequences converge and the limit of the denominator is nonzero, the limit of the quotient is equal to the quotient of the limits. This is essential in resolving expressions that are ratios of sequences, allowing direct substitution of the limits to evaluate the overall limit.

Sequence Convergence

This concept refers to the idea that a sequence approaches a specific value as its index goes to infinity. When a sequence converges, for any arbitrarily small positive number, there exists an index beyond which all the terms of the sequence lie within that small distance from the limit. This fundamental idea allows us to predict and use the behavior of sequences in limit problems.

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