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A sequence is considered monotonic if it is either entirely non-increasing or non-decreasing throughout its domain. This means that for a sequence \(a_n\), it is monotonic if the terms do not change their direction of progression; they either always go up, always go down, or stay the same as \(n\) increases.

For instance, if we have a sequence where each term is greater than or equal to the previous term (\(a_{n+1} \geq a_n\)), it's called a monotonically non-decreasing sequence. Conversely, a monotonically non-increasing sequence is one where each term is less than or equal to the previous one (\(a_{n+1} \leq a_n\)). These sequences have a certain predictability to their behavior making it easier to determine convergence. Monotonic sequences can also be bounded or unbounded which further aids in identifying their limits.

In contrast to monotonic sequences, an oscillating sequence is one that does not steadily increase or decrease but rather alternates or fluctuates between values. This means the terms of the sequence change signs or magnitudes in a pattern that does not maintain a singular direction.

Oscillating sequences, such as \( \left\{(-0.7)^n\right\} \), where \(n\) is a positive integer, are characterized by this swinging behavior between values. When \(n\) is even, \( (-0.7)^n \) is positive, and when \(n\) is odd, \( (-0.7)^n \) is negative. The oscillatory nature can make it more challenging to determine the sequence's behavior at infinity and thus its convergence or divergence. However, some oscillating sequences may still converge if the magnitude of their terms approaches a common value as \(n \) gets very large, which is the case for our example as both the positive and negative terms tend towards 0.

The limit of a sequence is a fundamental concept in calculus, particularly in the study of sequences and series. It is the value that the terms of a sequence approach as the index \(n\) becomes arbitrarily large. In symbolical terms, a sequence \(\left\{a_n\right\}\) has a limit \(\textbf{L}\) if for every positive number \(\varepsilon\), there is a corresponding positive integer \(N\) such that for all \(n > N\), the terms \(a_n\) satisfy the inequality \(\left| a_n - L \right| < \varepsilon\). In simpler terms, we're saying that after a certain point, all terms of the sequence get arbitrarily close to \(\textbf{L}\) and stay close.

For sequences that are both monotonic and bounded, the Monotone Convergence Theorem guarantees the existence of a limit. However, even some oscillating sequences may have limits if the absolute value of their terms decreases over time, leading them to converge to a particular value, as seen with \(\left\{(-0.7)^{n}\right\} \) converging to 0. Determining limits is crucial since it helps us understand the long-term behavior of sequences and the functions they may represent.