Limits of Sequences
Limits of Sequences
A sequence is a list of numbers arranged in a definite order, such as:
When we talk about the limit of a sequence, we are asking what happens to the terms as we move further and further along the list. If the terms settle closer and closer to a specific number, we say the sequence has a limit.
Intuitive Meaning of a Limit
Imagine a sequence where the terms keep getting closer to a fixed number. For example:
Looking at these terms, we see that they are approaching \( \frac{10}{9} \) (which is approximately 1.1111…). No matter how far we go, the values will never jump away from \( \frac{10}{9} \); instead, they will get as close as we want. If we continue indefinitely, the terms will look more and more like \( \frac{10}{9} \), though they may never exactly equal it.
When this happens, we write:
This means that as \(n\) becomes very large, the terms of the sequence get arbitrarily close to \( \frac{10}{9} \) and never drift away.
When a Sequence Has a Limit
A sequence has a limit if its terms get closer and closer to a single number and never wander off unpredictably. Consider the following cases:
- The sequence \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\) keeps decreasing and approaches 0. Here, we say:
[ \lim_{n \to \infty} \frac{1}{n} = 0. ]
- The sequence \(2, 4, 8, 16, \dots\) grows larger and larger without settling near any fixed number. This sequence does not have a finite limit; we say it diverges to infinity.
Convergent vs. Divergent Sequences
A sequence \(\{a_n\}\) can behave in two fundamentally different ways as \(n\) increases: it can settle near a single number (converge), or it can fail to do so (diverge).
Convergent Sequences
A sequence is called convergent if its terms get closer and closer to a fixed number as \(n\) increases. This fixed number is called the limit of the sequence.
Consider the sequence:
Observing the terms, we see that each one is slightly less than 1 and getting closer and closer to 0.99. No matter how far we go, the values will not jump away from 0.99; instead, they will continue approaching it.
Thus,
Other examples of convergent sequences:
- The sequence \( \frac{1}{n^2} \) (i.e., \(1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \dots\)) converges to 0.
- The sequence \( 1 + \frac{1}{n} \) (i.e., \(2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \dots\)) converges to 1.
A key property of convergent sequences is that their terms remain bounded and settle near a single value as \(n\) increases.
Divergent Sequences
A sequence is called divergent if it does not settle near a single number. This can happen in two ways:
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The sequence keeps increasing (or decreasing) indefinitely.
Consider the sequence:
\[ 2, 4, 6, 8, 10, \dots \]This is an arithmetic progression with common difference \(2\). The terms grow larger and larger without approaching any fixed number. The sequence diverges to infinity, which we write as:
\[ \lim_{n \to \infty} a_n = \infty. \]Similarly, the sequence \( -1, -3, -5, -7, \dots \) also diverges, but in the negative direction.
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The sequence oscillates and does not settle.
Consider:
\[ 1, -1, 1, -1, 1, -1, \dots \]This sequence keeps switching between 1 and \(-1\), never approaching a single number. It does not get arbitrarily close to any fixed value, so it diverges.
Boundedness and Convergence of Sequences
A fundamental property of convergent sequences is that they must be bounded. If a sequence \(\{a_n\}\) converges to some limit \(L\), then beyond a certain index, all terms remain arbitrarily close to \(L\), ensuring that the sequence does not diverge to infinity or oscillate unboundedly. Thus, there exists a real number \(M > 0\) such that \(|a_n| \leq M\) for all \(n\), proving that every convergent sequence is bounded.
However, the converse does not hold: a sequence can be bounded without being convergent. Consider the sequence
which oscillates between \(1\) and \(-1\) indefinitely. Clearly, it is bounded, as \(|a_n| \leq 1\) for all \(n\), yet it does not approach a single limit. This demonstrates that boundedness alone is insufficient to guarantee convergence.
A function is said to be continuous if its graph can be drawn without lifting the pen from the paper. This intuitive idea suggests that a continuous function does not exhibit sudden jumps, breaks, or holes. More precisely, continuity captures the idea that small changes in the input should lead to small changes in the output, ensuring that the function behaves predictably and smoothly.
This notion of predictability is fundamental in mathematical analysis. If a function is continuous at a point, then as the input approaches that point, the output of the function approaches the function’s value at that point. There is no sudden or arbitrary change in the function’s behavior. A discontinuous function, on the other hand, introduces an abrupt variation in value, indicating an essential failure of smoothness. For example, the function
exhibits a discontinuity at \( x = 1 \) because the limit from the left differs from the limit from the right, creating a break in the graph. Such a function does not allow a seamless transition from one side of \( x = 1 \) to the other, making the discontinuity evident.
This concept of continuity has strong physical and geometric significance. In real-world scenarios, continuity often corresponds to smooth and stable change. Consider the position of an object moving along a path. If the position of the object as a function of time is continuous, then it does not teleport from one location to another instantaneously. The motion occurs smoothly, without sudden jumps. However, if the function governing position were discontinuous, this would imply an instantaneous shift in location, which contradicts physical reality. Similarly, if a temperature function were discontinuous, it would mean a sudden and unphysical jump in temperature at some moment, which is inconsistent with how heat transfer operates in real materials.
Example
The position-time (\( x \)-\( t \)) graph illustrates an object moving along a straight trajectory, then abruptly teleporting to a new position at \( t = 2 \).
Initially, for \( t < 2 \), the object follows the linear path \( x = 2t \), increasing its position smoothly. At \( t = 2 \), it reaches \( x = 4 \), but instead of continuing along this path, it instantly teleports to \( x = 0 \). This discontinuity is marked by a hollow circle at \( (2,4) \) and a filled circle at \( (2,0) \), indicating that the object was at \( x = 4 \) just before \( t = 2 \) but reappeared at \( x = 0 \) immediately afterward.
After teleportation, for \( t \geq 2 \), the object resumes motion along a new trajectory \( x = 2t - 4 \), starting from \( x = 0 \). This suggests that after teleporting, the object follows a different linear path, maintaining continuity from its new position.
This graph effectively visualizes an instantaneous position shift—a phenomenon that is impossible under normal physical motion but can occur in theoretical models, simulations, or abstract mathematical constructs.
The idea of continuity thus ensures that a function reflects the natural behavior of many physical and geometric processes. A discontinuity in a function signals a fundamental disruption—either a physical impossibility or an abrupt transition in the modeled system. Hence, continuity is not merely a convenient mathematical property but a necessary condition for well-behaved, realistic models in both pure and applied mathematics.