Limits, Intuition, and One-Sided Concepts

Instantaneous velocity vs average velocity

• In motion problems, we often ask: how fast is something going?
• Average velocity between two times is the slope of the line connecting the two points on the position-vs-time graph.
  • Slope = rise over run = change in y over change in x.
• Instantaneous velocity at a point is the slope of the tangent line at that point.
  • If you only know an average over a longer interval, it may be misleading (e.g., average 51 mph vs actual 68 mph at a moment).
• To get an instantaneous quantity, we need a new mathematical idea: limits.

The need for a new mathematical idea: limits

• Limits address what a quantity approaches when observations get arbitrarily close to a point, even if we cannot observe directly at that point.
  • Example: predicting black-hole properties by looking at surrounding data, not by entering the hole.
  • Example: you don’t need data for everyone in a country to make a prediction about a population.
• In math, limits let us predict a y-value when x is near a, even if at x = a the function is not well-behaved (e.g., division by zero, holes).
• The limit is about the neighborhood around a, not about the actual value at a.

Book-style vs intuitive definitions

• The book defines a limit with precision: you can make the values of f(x) arbitrarily close to L by taking x sufficiently close to a.
• Intuition: the limit as x → a is the best prediction for f(x) based on nearby data, regardless of what f(a) actually is.
• Important caution: reading a math definition is not like reading ordinary text; every term carries meaning. Think in terms of neighborhoods, paths, and approaching values.

Formal and intuitive ideas of a limit

• Common notations and ideas:
  • f(x) can be a function, an equation, a table of values, or a graph (a curve through space).
  • a is the moment (often time) you're interested in; the function must be defined on an open interval containing a (i.e., on both sides of a).
  • The limit concerns x-values that are near a from both sides, not necessarily exactly at a.
  • $\lim_{x\to a} f(x) = L$ means as x gets arbitrarily close to a, the y-values f(x) get arbitrarily close to L.
  • The fact that the function may blow up exactly at a (hole or vertical asymptote) does not prevent a limit from existing.
• Important distinction: the limit describes the behavior near a, not the actual value at a.

One example to illustrate the concept visually

• Suppose a graph has a hole at x = a with f(a) undefined or a different value than the surrounding curve.
  • The limit as x → a is the value the y-coordinate would approach if you could plug in x values arbitrarily close to a (excluding a itself).
  • The actual value f(a) may be something else (or undefined).

A concrete two-point intuition about limits

• Example: consider a graph where as x approaches 4 along values not equal to 4, the y-values approach 3, but f(4) = 1.
  • Then: $\lim_{x\to 4} f(x) = 3,$ but $f(4) = 1.$
  • This illustrates the limit vs the function value distinction.

A piecewise function illustrating a limit that exists but a different value at the point

• Function G(x) defined as:
  • $G(x) = \frac{x-1}{x^2-1} \quad (x \neq 1),$
  • $G(1) = 2.$
• For x ≠ 1, simplify: $\frac{x-1}{x^2-1} = \frac{x-1}{(x-1)(x+1)} = \frac{1}{x+1}.$
• As x → 1, $\lim_{x\to 1} G(x) = \frac{1}{1+1} = \frac{1}{2}.$
• However, the actual value at x = 1 is G(1) = 2.
• This shows a limit existing independently of the function value at the point (discontinuity).

Notation: left-hand and right-hand limits

• Sometimes you can only access x-values from one side of a; this yields one-sided limits.
• The limit from the right (x > a) is denoted by:
  • $\lim\limits_{x\to a^+} f(x) = L,$ meaning as x approaches a from the right, f(x) → L.
• The limit from the left (x < a) is denoted by:
  • $\lim\limits_{x\to a^-} f(x) = L,$ meaning as x approaches a from the left, f(x) → L.
• If both one-sided limits exist and are equal (L from the left and the right), then the two-sided limit exists and equals that common value.
• If the one-sided limits disagree, the limit does not exist (DNE).

One-sided limit intuition via a quick example

• Imagine a graph where as x→a from the right, f(x) → 4, and as x→a from the left, f(x) → 3.
  • Then $\lim_{x\to a} f(x)$ does not exist (DNE) because the two sides disagree.
• If instead both sides approach the same value L, the two-sided limit exists and equals L.

The value of the function vs the limit

• Even when a limit exists, the function value at a may be different or undefined.
  • Example: a closed dot at x = a with y = L would mean the function value equals the limit, which is a sign of continuity at a.
  • An open circle at x = a indicates the point is not actually part of the function’s graph there; the limit can still exist.
• If left- and right-hand limits agree and the function is defined at a with that same value, the function is continuous at a.

A caveat: when the limit does not exist

• Reasons limits may not exist:
  • Left and right limits disagree (as described above).
  • Rapid or infinite oscillation near a (the function keeps zig-zagging faster and faster).
  • A vertical asymptote or unbounded behavior prevents approaching a finite L.
• Example of nonexistence due to oscillation: f(x) = \sin(\pi/x) as x → 0.
  • The values of f(x) oscillate between -1 and 1 infinitely often as x → 0, so there is no single limit L.
  • In limit-land, you cannot assign a single y-value that all nearby x-values approach.

A classic oscillation example related to limits

• Consider the function f(x) = \sin(\pi/x) as x → 0.
  • The limit does not exist because the function oscillates endlessly between -1 and 1 near 0.
  • This is an example of hyperoscillation and demonstrates why limits are needed to formalize what it means for a function to have a predictable behavior near a point.

Summary of key ideas

• The limit describes the behavior of f(x) near a, not necessarily at a itself.
• The limit can exist even if f(a) is undefined or not equal to the limit value.
• One-sided limits help when data or graphs are only available from one side of a point.
• When left and right limits agree, the limit exists; when they disagree, the limit does not exist.
• Open circles and closed dots distinguish points that are not included in the function versus points where the function is defined.
• Some functions can have limits but be discontinuous at the point where the limit applies (discontinuities can be mild or more exotic).
• Hyperoscillations provide a cautionary counterexample where limits fail to exist despite approachable neighborhoods.

Formal definitions and a concise epsilon-delta form

• Intuitive book-style statement: For every neighborhood around L, there exists a neighborhood around a such that f(x) lies in that neighborhood for all x sufficiently close to a (x ≠ a).
• Formal (epsilon-delta) definition:
  \lim_{x\to a} f(x) = L \quad \text{iff} \quad \forall \varepsilon>0\ \exists \delta>0\ \text{such that } 0<|x-a|<\delta \Rightarrow |f(x)-L|<\varepsilon.

Quick reference: practical takeaway for limits

• To determine a limit, look at values of f(x) as x gets arbitrarily close to a (from both sides if possible).
• If the surrounding values approach a single number L, the limit is L.
• If approaching from one side only, use the appropriate one-sided limit notation.
• If the left and right approaches disagree, the limit does not exist.
• The value at a may or may not equal the limit; when they match (and the function is defined there), the function is continuous at a.
• Algebraic manipulation, factoring, and canceling common factors can reveal limits; numerical plugging is often just a rough check.

A note on derivatives (instantaneous rate of change)

• Conceptual link: instantaneous velocity is the limit of average velocity as the time interval shrinks to zero, i.e., the slope of the tangent line at a point on the position curve.
• Standard derivative formula (as a consistent next step):
  $f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}.$
• The derivative is the slope of the tangent line at x = a.

Additional examples you might encounter on exams

• Example: A graph with a hole at x = a and a different defined value at a.
  • Limit as x → a exists and equals L, but f(a) ≠ L or f(a) is undefined.
• Example: A piecewise function where for x ≠ a, f(x) follows one expression, and at x = a, f(a) is set to a different value or left undefined.
  • The limit can still exist if both sides approach the same L.
• Example: If left-hand limit = right-hand limit = L, then $\lim_{x\to a} f(x) = L.$ If f(a) is undefined or f(a) ≠ L, the function is still said to have a limit at a but is not continuous there.

Summary checklist for limits (quick study aid)

• Is f(x) approaching a single value as x → a? If yes, identify that value as the limit.
• Do left and right limits agree? If not, DNE.
• Is f(a) defined? If yes, does f(a) equal the limit? If yes, the function is continuous at a.
• Can the limit be computed by simplifying, factoring, or by known limit laws? Use algebraic techniques where possible.
• If the function oscilliates infinitely fast near a, expect DNE (hyperoscillation like \sin(\pi/x) near x = 0).

Formulas to remember

• Two-sided limit: $\lim_{x\to a} f(x) = L.$
• One-sided limits: $\lim<em>{x\to a^+} f(x) = L\quad\text{and}\quad \lim</em>{x\to a^-} f(x) = L.$
• Formal epsilon-delta definition: \forall \varepsilon>0\ \exists \delta>0\ \text{such that } 0<|x-a|<\delta \Rightarrow |f(x)-L|<\varepsilon.
• Derivative as a limit: $f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}.$