Notes on Section 2.2: The Limit of a Function

The Limit of a Function (Section 2.2) – Comprehensive Notes

Context and purpose
- Limits describe the behavior of f(x) as x gets arbitrarily close to a, not necessarily at x = a or even where f is defined at a.
- This section introduces intuitive and formal definitions, numerical/graphical approaches, one-sided limits, infinite limits, vertical asymptotes, and common pitfalls.
Basic intuitive definition of a limit
- If f is defined on some open interval that contains a (except possibly at a), we say:
- $\lim_{x \to a} f(x) = L$
- meaning: we can make f(x) as close to L as we like by taking x sufficiently close to a, from either side, but with x ≠ a.
- The value at a, i.e., f(a), need not equal L and f(a) need not even be defined.
- Alternative (equivalent) notation:
- $f(x) \to L \quad \text{as } x \to a$
- The emphasis is that x approaches a from the left and/or right, but x = a is excluded.
Example: a simple limit as x approaches 1
- For the function f(x) = \frac{x-1}{x^2-1}, as x → 1 the function values approach 0.5.
- The limit is:
- $\lim_{x \to 1} \frac{x-1}{x^2-1} = \frac{1}{2}.$
- This illustrates that the limit can exist even if the function is not defined at a (or if the simplification would require factoring/rewriting).
The role of graphs and numerical tables
- Numerical tables and graphs near a can suggest the limit value, but rounding and finite precision can mislead (especially when a is involved in expressions like square roots, etc.).
- Example discussion (from the transcript): values of a function near a can appear to approach a certain number, but calculators/graphs may introduce rounding errors that misrepresent the true limit.
One-sided limits (Definition 2)
- Left-hand limit:
- $\lim_{x \to a^-} f(x) = L$
- Right-hand limit:
- $\lim_{x \to a^+} f(x) = L$
- The traditional (two-sided) limit exists if and only if both one-sided limits exist and are equal:
- $\lim{x \to a} f(x) = L \quad \text{iff} \quad \lim{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L.$
Example: Heaviside function (H)
- Heaviside function H(t) is defined by
- $H(t) = \begin{cases} 0, & t < 0, \ 1, & t > 0. \end{cases}$
- (Some definitions specify a value at t = 0; the graph typically shows a jump discontinuity at t = 0.)
- There is no single value of lim_{t \to 0} H(t) because the left-hand limit is 0 and the right-hand limit is 1:
- $\lim{t \to 0^-} H(t) = 0, \quad \lim{t \to 0^+} H(t) = 1.$
- This motivates the distinction between left- and right-hand limits and shows that a two-sided limit may fail to exist when one-sided limits disagree.
Definitions and notational conventions for limits
- Two-sided limit definition emphasizes that x approaches a from either side, with x ≠ a.
- One-sided limits are used to describe behavior from only one side and can explain vertical asymptotes and discontinuities.
Example 4 (graph-based limits)
- Given a graph g(x) with a = 2:
- Left-hand limit: as x → 2⁻, g(x) → 3.
- Right-hand limit: as x → 2⁺, g(x) → 1.
- Since the two one-sided limits are not equal, the two-sided limit does not exist:
- $\lim_{x \to 2} g(x) \text{ does not exist}.$
- At x = 5, both one-sided limits exist and equal 2, so
- $\lim_{x \to 5} g(x) = 2,$ but note that g(5) might not equal this limit value.
Infinite limits and vertical asymptotes (Definitions 4–6 in the chapter)
- Infinite limit (two-sided):
- If f is defined on both sides of a, except possibly at a, and as x → a,
- $\lim_{x \to a} f(x) = \infty$
- means f(x) can be made arbitrarily large by taking x sufficiently close to a (but x ≠ a). The symbol ∞ is not a number.
- Similarly,
- $\lim_{x \to a} f(x) = -\infty$
- means f(x) can be made arbitrarily large in magnitude and negative.
- One-sided infinite limits: define similarly for x → a⁻ or x → a⁺.
- Vertical asymptote: the vertical line x = a is a vertical asymptote of y = f(x) if any of the (one-sided) limits above occur:
- $\lim{x \to a^-} f(x) = \infty,\quad \lim{x \to a^+} f(x) = \infty,\quad \lim{x \to a^-} f(x) = -\infty,\quad \lim{x \to a^+} f(x) = -\infty.$
- Example: f(x) = 1/x^2 has a vertical asymptote at x = 0 because as x → 0, f(x) → +∞ from both sides.
- Practical interpretation: vertical asymptotes help in sketching graphs and understanding unbounded behavior near certain x-values.
Examples illustrating limits and their failures
- Example 5: lim_{x \to 0} sin(π/x) does not exist because sin(π/x) oscillates between -1 and 1 infinitely often as x → 0, so the function values do not approach any fixed L.
- Example 6: lim_{x \to 0} 1/x^2 does not exist as a finite number; however, we describe the behavior as an infinite limit:
- $\lim_{x \to 0} \frac{1}{x^2} = \infty.$
- This is an example of a vertical asymptote at x = 0.
Infinite limits and notation examples
- To emphasize the nature of divergence to infinity, we may write:
- $\lim{x \to a} f(x) = \infty \qquad \text{or} \qquad \lim{x \to a} f(x) = -\infty.$
- It is common to also write the equivalent one-sided forms when describing behavior near a vertical asymptote.
Example 7: Does the curve y = \frac{2x}{x-3} have a vertical asymptote?
- Denominator vanishes at x = 3, so potential vertical asymptote at x = 3.
- Examine one-sided limits:
- As x → 3⁺, the denominator is a small positive number, numerator ~ 6, so f(x) → +∞.
- As x → 3⁻, the denominator is a small negative number, numerator ~ 6, so f(x) → -∞.
- Therefore, x = 3 is a vertical asymptote for y = \frac{2x}{x-3}.
Example 8: Vertical asymptotes for common functions
- Tangent: f(x) = \tan x = \frac{\sin x}{\cos x}
- Potential vertical asymptotes where cos x = 0, i.e., at x = \frac{\pi}{2} + k\pi for any integer k.
- Indeed, as x → (\pi/2)⁺, tan x → +∞; as x → (\pi/2)⁻, tan x → -∞.
- Natural logarithm: f(x) = \ln x
- As x → 0⁺, ln x → -∞, so x = 0 is a vertical asymptote (for the graph on x > 0).
- Logarithm with base b > 1 behaves similarly with a vertical asymptote at x = 0.
- Visual check via graphs confirms these asymptotes.
Practical takeaways
- Limits describe local behavior near a; they do not always depend on the value at a.
- One-sided limits are essential for understanding discontinuities and asymptotic behavior.
- Infinite limits and vertical asymptotes model quantities that grow without bound as x approaches a.
- Graphs and numerical experiments can mislead due to rounding errors; careful algebraic or analytical methods (where possible) yield foolproof results.
Limits and technology (brief discussion)
- Computer algebra systems (CAS) and numerical tools can compute limits, but they may encounter pitfalls if relying solely on numerical experimentation.
- In practice, limit calculation often relies on algebraic manipulation, series expansions, or geometric arguments (as discussed for specific limits like sin x / x).
- The chapter emphasizes developing foolproof methods rather than trusting raw numerical tables or graphs alone.
Connections to broader themes
- Limits underpin derivative definitions (tangent slope) and velocity concepts, tying Section 2.2 to later topics in differentiation.
- Understanding limits is foundational for continuity, infinite behavior, and asymptotic analysis used in real-world modeling.
Formulas and key results (quick reference)
- Limit notation and existence
- $\lim_{x \to a} f(x) = L$
- One-sided limits:
- $\lim{x \to a^-} f(x) = L, \quad \lim{x \to a^+} f(x) = L$
- If both one-sided limits exist and are equal, the two-sided limit exists and equals L.
- Heaviside function example:
- $H(t) = \begin{cases} 0, & t < 0, \ 1, & t > 0, \end{cases}$
- One-sided limits can differ at a jump discontinuity (as with H at t = 0).
- Infinite limits and vertical asymptotes:
- \lim_{x \to a} f(x) = \infty \,\/\, -\infty
- Vertical asymptote: line x = a where f(x) grows without bound as x approaches a from one or both sides.
- Example results:
- $\lim_{x \to 0} \frac{1}{x^2} = \infty.$
- \lim{x \to 3^+} \frac{2x}{x-3} = +\infty,\quad \lim{x \to 3^-} \frac{2x}{x-3} = -\infty.$n
- Example limits discussed in the section:
- \lim_{t \to 0} \frac{\sqrt{t^2+9} - 3}{t^2} = \frac{1}{6}. $</li> <li>$ \lim_{x \to 0} \frac{\sin x}{x} = 1. $</li> <li>$ \lim_{x \to 0} \frac{x^3 + \cos(5x)}{10000} = 0.0001. $</li> <li>Nonexistence examples:</li> <li>$ \lim_{x \to 0} \sin\left(\frac{\pi}{x}\right) \quad \text{does not exist}.$$
- Graphical/Illustrative figures accompany these discussions in the text.
Summary of essential ideas for exam preparation
- Be able to articulate the intuitive definition of a limit and why it cares about x near a, not at a.
- Distinguish between two-sided and one-sided limits; know when a limit exists based on one-sided limits.
- Recognize and describe infinite limits and vertical asymptotes, and identify the x-values where they occur from the function (e.g., poles, discontinuities).
- Use examples to illustrate potential pitfalls in numerical/graphical estimation (rounding, oscillation, etc.).
- Remember key limits and their general ideas, such as sin x / x → 1 and expansions leading to limits like 1/6 in specific conjugate-rationalizations.
Quick reference table of topics covered
- Limit existence and notation: two-sided and one-sided limits
- Heaviside function as an example of nonexistence of a two-sided limit at 0
- Infinite limits and vertical asymptotes definitions and examples
- Examples illustrating correct reasoning versus numerical/graphical pitfalls
- The role of technology in limit calculation and the importance of analytical methods