2 LIMITS

• Limits form the foundation for calculus and enable solutions to fundamental problems about rates of change and areas.
  The chapter establishes that understanding limits is essential before developing the main tools and applications of calculus.
• Einstein's theory of relativity imposes an upper speed limit on objects in the universe - the speed of light - which cannot be exceeded.
  This physical example illustrates the concept of a limit as an absolute boundary or constraint that cannot be transcended, motivating the mathematical study of limits.

Source: Introduction, pages 1-2

• The tangent line problem asks for the slope of a line that touches a curve at exactly one point, which requires understanding the rate of change of a function at a specific location.
  Finding the slope of a tangent line is not straightforward using standard formulas because you cannot select two distinct points on a curve to compute slope when the tangent touches at only one point. This problem motivates the need for limits.
• The area problem seeks to find the area of a region bounded by a curved boundary, which cannot be calculated using basic geometric formulas.
  Traditional area formulas work for polygons and circles but fail for regions with curved edges. Calculus provides a systematic method to approximate and then precisely calculate these areas through limits.
• Limits are the fundamental concept that resolves both the tangent line and area problems in calculus.
  The tangent problem uses limits to find instantaneous rates of change by computing slopes of secant lines as two points approach each other. The area problem uses limits to sum infinitesimal rectangular strips whose widths approach zero.
• A secant line passes through two points on a curve, and as one point slides toward the other, the secant line approaches the tangent line.
  This approximation process depends on the behavior of the secant slope as the second point gets arbitrarily close to the first, which is precisely what a limit describes.

Source: Section 2.1, pages 2-12

• The limit of a function f(x) as x approaches a value a is L if f(x) gets arbitrarily close to L as x gets arbitrarily close to a, written as lim(x→a) f(x) = L.
  This intuitive definition means the output values can be made as close to L as desired by choosing input values sufficiently close to a. The limit concerns what f approaches, not the actual value f(a).
• Limits can be determined by constructing a table of function values as x approaches the target value from both directions.
  Observing the pattern of output values in the table reveals the y-coordinate that the function approaches, providing an empirical method to conjecture the limit's value.
• A limit from the right, written lim(x→a+) f(x) = L, represents the behavior of f(x) as x approaches a from values greater than a.
  This one-sided limit allows analysis of function behavior on a particular side of a point, which is essential when the two-sided limit may not exist or when the function behaves differently on each side.
• A limit from the left, written lim(x→a-) f(x) = L, represents the behavior of f(x) as x approaches a from values less than a.
  For a two-sided limit to exist, both the left-hand and right-hand limits must exist and be equal; if they differ, the two-sided limit does not exist.
• An infinite limit occurs when f(x) increases or decreases without bound as x approaches a, written as lim(x→a) f(x) = infinity or negative infinity.
  A vertical asymptote at x = a indicates an infinite limit; the function values grow arbitrarily large in absolute value near the point, though the limit value itself is not a finite number.
• A function value f(a) may equal the limit lim(x→a) f(x), but the existence of the limit does not depend on whether f is defined at x = a or what value it takes there.
  The limit describes what the function approaches as x nears a, which is determined by nearby points but independent of the point itself.

Source: Section 2.2, pages 12-36

• If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) [f(x) + g(x)] = L + M, and the limit of a sum equals the sum of the limits.
  This sum law, along with corresponding laws for differences, products, and quotients, forms the foundation for evaluating limits algebraically rather than numerically.
• The limit of a constant times a function equals the constant times the limit of that function: lim(x→a) [c*f(x)] = c*lim(x→a) f(x).
  This scalar multiplication law allows you to factor out constants when computing limits, simplifying many calculations.
• The limit of a power obeys lim(x→a) [f(x)]^n = [lim(x→a) f(x)]^n for positive integers n, allowing exponents to be applied after taking the limit.
  This power law is particularly useful for polynomial and rational functions, where you can evaluate the limit by substitution once the underlying function limits exist.
• For rational functions, if the denominator does not approach zero, you can evaluate the limit by direct substitution of x = a into the function.
  Direct substitution works when the function is continuous at the point, which is the case for polynomials and rational functions away from their poles.
• When direct substitution yields the indeterminate form 0/0, you must algebraically manipulate the expression by factoring or using other techniques to find the limit.
  Common techniques include canceling common factors in the numerator and denominator, or rationalizing numerators or denominators containing radicals.
• The squeeze theorem states that if f(x) ≤ g(x) ≤ h(x) for all x near a, and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L.
  This theorem is essential for evaluating limits that cannot be computed algebraically, such as lim(x→0) [x*sin(1/x)] = 0, by bounding g between two functions with the same limit.
• The limit of the sine function as x approaches 0 is lim(x→0) sin(x) = 0, and the special limit lim(x→0) sin(x)/x = 1 is fundamental in calculus.
  These sine limits, particularly the ratio sin(x)/x as x approaches 0, appear frequently in derivatives and are proven using the squeeze theorem and geometric arguments.

Source: Section 2.3, pages 36-51

• A function f is continuous at a point a if lim(x→a) f(x) = f(a), which requires three conditions: f(a) is defined, the limit exists, and the limit equals the function value.
  Continuity combines the existence of a limit with agreement between the limit and the actual function value at that point. This is the foundation for determining whether a function behaves smoothly without jumps or breaks.
• A function is continuous on an interval if it is continuous at every point in that interval.
  This global definition extends the point-wise concept of continuity across an entire domain or region where the function operates.
• Discontinuities are classified as removable (where the limit exists but differs from or is undefined at the point), jump (where left and right limits exist but differ), or infinite (where the limit approaches infinity).
  Understanding discontinuity types helps identify the nature of breaks in a function and whether they can theoretically be 'fixed' by redefining a single point.
• All polynomial functions are continuous everywhere, and rational functions are continuous at every point in their domain.
  These are fundamental continuous functions; polynomials have no breaks, and rational functions are continuous except where the denominator equals zero.
• The Intermediate Value Theorem states that if f is continuous on a closed interval [a,b] and N is any value between f(a) and f(b), then there exists some c in (a,b) where f(c) = N.
  This theorem guarantees that a continuous function must pass through every y-value between its endpoints, which is useful for proving the existence of solutions to equations.
• Sums, differences, products, quotients (where denominators are nonzero), and compositions of continuous functions are continuous.
  These limit laws translate to continuity, allowing you to build complex continuous functions from simpler continuous pieces.

Source: Section 2.4, pages 51-65

• The epsilon-delta definition formalizes the intuitive notion of a limit using precise mathematical language: the limit of f(x) as x approaches a equals L if for every epsilon > 0, there exists a delta > 0 such that whenever 0 < |x - a| < delta, then |f(x) - L| < epsilon.
  This definition quantifies what it means for functional values to approach L: no matter how small a distance epsilon we specify around L, we can always find a small enough distance delta around a such that all x-values within delta of a (except possibly a itself) produce f(x)-values within epsilon of L. The definition translates abstract closeness into rigorous inequalities.
• The precise definition can be understood geometrically: as epsilon (distance from the limit) decreases, delta (distance from the point a) also becomes smaller, but a delta always exists that works for any chosen epsilon.
  Figures 2.39 and 2.40 demonstrate how for progressively smaller epsilon values, we can find corresponding delta values that ensure the function stays within the epsilon band around L whenever x is within the delta band around a. This geometric visualization confirms the logical structure of the definition.
• To prove lim[x->a] f(x) = L, one must show that for any positive epsilon, a suitable positive delta can be found such that the conditional statement 'if |x - a| < delta then |f(x) - L| < epsilon' holds.
  The standard proof structure begins by letting epsilon be arbitrary, then chooses delta (often by algebraic or geometric manipulation of the inequality |f(x) - L| < epsilon), assumes |x - a| < delta, and derives |f(x) - L| < epsilon. The choice of delta may depend on epsilon but must not depend on x.
• For linear functions, finding delta is straightforward algebraically by manipulating the target inequality |f(x) - L| < epsilon to isolate |x - a|, as demonstrated in Examples 2.39 and 2.40.
  For instance, to prove lim[x->2] (3x - 2) = 4, one manipulates |3x - 6| < epsilon to get |x - 2| < epsilon/3, so choosing delta = epsilon/3 works. This algebraic approach is direct for simple functions.
• For nonlinear functions, a geometric approach or careful algebraic handling is needed, often requiring an upper bound on epsilon to ensure delta exists as a function of epsilon only (Example 2.41).
  When the relationship between |f(x) - L| and |x - a| is complex, assuming epsilon is smaller than some fixed positive value (such as epsilon < 1) can simplify the algebra and allow delta to be expressed solely in terms of epsilon, satisfying the definition's requirement.
• The triangle inequality is a key tool in proving limit laws: |a + b| <= |a| + |b| for any real numbers a and b.
  This inequality allows one to bound the absolute value of a sum by the sum of absolute values, which is essential when proving statements like the sum law for limits where the combined error must be controlled.
• A limit does not exist if for every candidate value L, there exists some epsilon > 0 such that for all delta > 0, there is an x with 0 < |x - a| < delta but |f(x) - L| >= epsilon.
  This negation of the epsilon-delta definition means we must find at least one epsilon for which no delta works; this is demonstrated in Example 2.43 for the floor function, where oscillating behavior prevents any single L from satisfying the limit definition.
• The formal epsilon-delta definitions of one-sided limits modify the standard definition by replacing 0 < |x - a| < delta with a < x < a + delta (right limit) or a - delta < x < a (left limit).
  These definitions restrict consideration to x-values on one side of a only, allowing precise treatment of limits from the right and left, which are essential when a function behaves differently on either side of a point.
• For infinite limits, the definition replaces the requirement |f(x) - L| < epsilon with f(x) > M (for positive infinity) or f(x) < -M (for negative infinity), where M is an arbitrarily large positive number.
  The definition lim[x->a] f(x) = infinity means: for every M > 0, there exists delta > 0 such that if 0 < |x - a| < delta then f(x) > M. This formalizes the idea of function values becoming arbitrarily large as x approaches a.
• The epsilon-delta definition provides the rigorous foundation needed to prove the limit laws (sum law, product law, quotient law, etc.) that were stated intuitively in earlier sections.
  By working directly from the definition, one can prove that if lim[x->a] f(x) = L and lim[x->a] g(x) = M, then lim[x->a] [f(x) + g(x)] = L + M, and similarly for other operations, ensuring the limit laws are mathematically sound.

Source: Section 2.5, pages 65-77

• A function is continuous at a point a if and only if f(a) is defined, the limit as x approaches a exists, and the limit equals f(a).
  These three conditions must all be satisfied simultaneously for continuity at a point. If any condition fails, the function is discontinuous at that point.
• Discontinuities are classified into three types: removable (limit exists but function undefined or unequal to limit), jump (left and right limits exist but are unequal), and infinite (limit approaches infinity).
  Understanding these classifications helps identify the nature of where a function breaks down and whether the discontinuity can be 'fixed' by redefining the function value.
• The Intermediate Value Theorem states that if f is continuous over a closed interval [a,b] and z is any value between f(a) and f(b), then there exists a point c in [a,b] where f(c) = z.
  This theorem guarantees that a continuous function takes on all intermediate values between its endpoints, which is useful for proving that solutions to equations exist.
• The epsilon-delta definition formalizes limits: the limit of f(x) as x approaches a equals L if for every epsilon > 0 there exists delta > 0 such that if 0 < |x - a| < delta then |f(x) - L| < epsilon.
  This rigorous definition converts the intuitive notion of limits into a precise mathematical statement that can be used to prove limit statements.
• Limit laws allow direct evaluation of limits of polynomials and rational functions by substitution without step-by-step processes, provided the denominator is nonzero.
  The limit laws (sum, product, quotient, power, root) provide shortcuts for calculating limits algebraically rather than using tables or graphs.
• The squeeze theorem states that if g(x) is less than or equal to f(x) which is less than or equal to h(x) over an interval containing a, and if the limits of g and h as x approaches a both equal L, then the limit of f as x approaches a also equals L.
  This theorem is valuable for finding limits of functions that are difficult to evaluate directly but can be bounded between two simpler functions.
• A tangent line to a curve at a point is the line that secant lines approach as their second points move along the curve toward the point of tangency.
  The slope of the tangent line measures the instantaneous rate of change of the function, which is the derivative and is found by evaluating a limit of secant line slopes.
• Differential calculus is built on the tangent problem and requires finding limits to calculate derivatives, while integral calculus is built on the area problem and also relies on limits.
  Both major branches of calculus fundamentally depend on the concept of limits, making limits the foundational tool for all calculus applications.
• A function has a vertical asymptote at x = a if the limit as x approaches a from the right or left is infinite.
  Vertical asymptotes indicate points where a function grows without bound and represent infinite discontinuities.
• One-sided limits describe the behavior of a function as it approaches a point from either the left or the right, and the two-sided limit exists only if both one-sided limits exist and are equal.
  One-sided limits are essential for understanding functions with jump discontinuities and for studying limits at endpoints of intervals.

Source: Chapter Review, pages 77-82