Theorem 1 states that if limx →∞ f(x)=L, then the sequence an= f(n) converges and limn →∞ an=L .

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Theorem 1 states that if limx →∞ f(x)=L, then the sequence...

Theorem 1 states that if $\lim _{x \rightarrow \infty} f(x)=L,$ then the sequence $a_{n}=$ $f(n)$ converges and $\lim _{n \rightarrow \infty} a_{n}=L .$ Show that the converse is false. In other words, find a function $f(x)$ such that $a_{n}=f(n)$ converges but $\lim _{x \rightarrow \infty} f(x)$ does not exist.

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Key Concepts

Limit at Infinity

This concept involves studying the behavior of a function as its input becomes arbitrarily large. When we say the limit of f(x) as x approaches infinity exists and equals L, we mean that for any given margin of error, there is a threshold beyond which all values of f(x) lie within that margin of L, irrespective of how x increases.

Sequence Convergence via Discrete Sampling

A sequence created by sampling a function at discrete points, such as taking a_n = f(n), may converge even if the overall function does not have a limit as its argument tends to infinity. This disparity arises because the discrete set of points can 'avoid' erratic or oscillatory behavior that the function may exhibit on intervals between those points.

Oscillatory Behavior

A function might oscillate or behave erratically between specific discrete values, preventing the existence of a limit as x approaches infinity. However, if, for example, the function’s oscillatory parts cancel out or consistently yield the same value at the sampled points, the resulting sequence may converge without the function having a well-defined limit over the entire domain.

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