Unit 2: Limits and Continuity

The concept of limits and continuity is one of the most crucial topics in calculus. These concepts are fundamental and are used to define all basic definitions of calculus, including continuous functions. While a conceptual definition might be sufficient for precalculus and introductory calculus, a technical explanation is required for higher levels.

Precise Definition of Limit

A limit is defined as a number that a function approaches as its independent variable approaches a particular value. For example, for the function f(x) = 4x, the limit of f(x) as x approaches 2 is 8, written symbolically as limx&to;2 (4x) = 8.

More precisely, a function f has a limit L as x approaches c if, given any positive number ε, there exists a positive number δ such that for all x, if 0 < |x - c| < δ, then |f(x) - L| < ε. This is written as limx&to;c f(x) = L. The existence of a limit as x approaches c does not depend on how the function may or may not be defined at c. For instance, a function f can have a limit of 2 as x approaches 1 even if f is not defined at 1, or if f(1) is not equal to 2. Similarly, the function f(x) = (sin x)/x approaches 1 as x approaches 0, even though f(0) is undefined.

Limits can be classified as one-sided or two-sided:

• The left-hand limit (limx&to;a− f(x)) is the expected value of f at x = a, considering values of 'f' near x to the left of 'a'.
• The right-hand limit (limx&to;a+ f(x)) is the expected value of f at x = a, considering values of 'f' near x to the right of 'a'.
• A two-sided limit (limx&to;a f(x)) exists if and only if both the right-hand and left-hand limits at 'a' exist and are equal. If they coincide, their common value is the limit of f(x) at x = a. For example, the greatest integer function, f(x) = int x, does not have a limit at integers because its left-hand and right-hand limits differ (e.g., at x=3, limx&to;3+ int x = 3 while limx&to;3− int x = 2).

Properties of Limits (for limx&to;c f(x) = L and limx&to;c g(x) = M):

• Sum Rule: The limit of the sum of two functions is the sum of their limits: limx&to;c [f(x) + g(x)] = L + M.
• Difference Rule: The limit of the difference of two functions is the difference of their limits: limx&to;c [f(x) - g(x)] = L - M.
• Product Rule: The limit of a product of two functions is the product of their limits: limx&to;c [f(x) • g(x)] = L • M.
• Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function: limx&to;c [k • f(x)] = k • L (where k is a constant).
• Quotient Rule: The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero: limx&to;c [f(x) / g(x)] = L / M, provided M ≠ 0.
• Power Rule: If r and s are integers and s ≠ 0, then limx&to;c [f(x)]r/s = Lr/s, provided Lr/s is a real number.
• The limit of any constant function is the constant term itself: limx&to;a C = C.
• The limit of the identity function is limx&to;c x = c.

Limits of polynomial functions and most rational functions can often be found by direct substitution (e.g., limx&to;c (x3 - 4x2 + 3) = c3 - 4c2 + 3), provided the denominator of a rational function is not zero. The Sandwich Theorem (Theorem 4) can also be used to find limits indirectly when direct calculation is difficult. If a function f is "sandwiched" between two other functions, g and h, such that g(x) ≤ f(x) ≤ h(x), and both g and h have the same limit L as x approaches c, then f also has that limit L. An example includes showing limx&to;0 (x2 sin(1/x)) = 0.

Limits at Infinity

Limits at infinity describe the behavior of a function when the values in its domain outgrow all finite bounds, either positively (x &to; ∞) or negatively (x &to; -∞).

• limx&to;∞ f(x) refers to the limit of f as x moves increasingly far to the right on the number line.
• limx&to;−∞ f(x) refers to the limit of f as x moves increasingly far to the left on the number line.

For example, limx&to;∞ (1/x) = 0 and limx&to;−∞ (1/x) = 0. The properties of limits (Sum, Difference, Product, Constant Multiple, Quotient, and Power Rules) also apply to limits as x approaches positive or negative infinity. The Sandwich Theorem also holds for limits as x approaches infinity; for instance, limx&to;∞ (sin x)/x = 0.

For numerically large values of x, the behavior of a complicated function can sometimes be modeled by a simpler "end behavior model" function. A function `g` is a right end behavior model for `f` if limx&to;∞ (f(x)/g(x)) = 1. Similarly, it's a left end behavior model if limx&to;−∞ (f(x)/g(x)) = 1. For a polynomial function f(x) = anxn + ... + a0, the term g(x) = anxn is an end behavior model. For rational functions, the ratio of the highest-degree terms in the numerator and denominator often serves as an end behavior model.

Continuity

Continuity is a central concept in calculus. Intuitively, a function is continuous at a particular point if its graph can be traced with a pencil without lifting the pencil from the paper's surface, meaning there is no break in its graph at that point.

A precise definition of continuity at a point is given using the idea of a limit:

• For an interior point c of its domain, a function y = f(x) is continuous at c if limx&to;c f(x) = f(c).
• For endpoints, it is continuous at a left endpoint a if limx&to;a+ f(x) = f(a), and at a right endpoint b if limx&to;b− f(x) = f(b).

For a function to be continuous at a specific point 'a', three conditions must be satisfied:

1. f(a) is defined.
2. limx&to;a f(x) exists.
3. limx&to;a+ f(x) = limx&to;a− f(x) = f(a) (the left-hand limit, right-hand limit, and the function's value at 'a' are all equal).

A continuous function is one that is continuous at every point in its domain. Examples include polynomial functions, rational functions (continuous at every point of their domains), the absolute value function, exponential functions, logarithmic functions, trigonometric functions, and radical functions (e.g., y = √nx where n > 1).

Properties of Continuous Functions (Theorem 6): If functions f and g are continuous at x = c, then their sums (f + g), differences (f - g), products (f • g), constant multiples (k • f), and quotients (f/g, provided g(c) ≠ 0) are also continuous at x = c.

Composite of Continuous Functions (Theorem 7): If f is continuous at c and g is continuous at f(c), then the composite function g(f(x)) is continuous at c. Examples include y = sin(x2) and y = |cos x|.

Intermediate Value Theorem for Continuous Functions (Theorem 8): If a function y = f(x) is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b). This means if y0 is between f(a) and f(b), there exists some c in [a, b] such that y0 = f(c). This theorem implies that the graph of a continuous function on an interval will be a single, unbroken curve, without any breaks or jumps.

Types of Discontinuity: A function is discontinuous when it has any gaps or breaks.

• Infinite Discontinuity (Asymptotic Discontinuity): Occurs where a vertical asymptote is present at x = a and f(a) is undefined. The function values approach positive or negative infinity on either side of the asymptote. Example: f(x) = 1/x2 at x = 0.
• Jump Discontinuity (Simple Discontinuity / Discontinuity of the First Kind): Occurs when the left-hand limit and the right-hand limit at a point exist but are not equal, although both limits are finite. Example: The greatest integer function at integer points.
• Removable Discontinuity (Positive Discontinuity): Occurs when a function has a predefined two-sided limit at x = a, but f(a) is either undefined at a or its value is not equal to the limit at a. This discontinuity can often be "removed" by redefining the function's value at that point to match the limit.
• Oscillating Discontinuity: Occurs when a function oscillates and has no limit as x approaches a point. Example: y = sin(1/x) at x = 0.

Horizontal Asymptotes

The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either the limit of f(x) as x approaches positive infinity is b (limx&to;∞ f(x) = b) or the limit of f(x) as x approaches negative infinity is b (limx&to;−∞ f(x) = b). For example, y = 0 is a horizontal asymptote for f(x) = 1/x. A function can have more than one horizontal asymptote; for instance, the function f(x) = x/√(x2+1) has two horizontal asymptotes, y = 1 and y = -1. Horizontal asymptotes are closely related to the end behavior models of functions.

Vertical Asymptotes

The line x = a is a vertical asymptote of the graph of a function y = f(x) if the function's values outgrow all positive or negative bounds as x approaches the finite number a. Specifically, this occurs if limx&to;a+ f(x) = ±∞ or limx&to;a− f(x) = ±∞. For example, x = 0 is a vertical asymptote for f(x) = 1/x and f(x) = 1/x2. Trigonometric functions like f(x) = tan x have infinitely many vertical asymptotes at values of x where cos x = 0 (e.g., odd multiples of π/2). It's important to note that a quotient function does not always have a vertical asymptote where its denominator is zero; for instance, limx&to;0 (sin x)/x = 1, so there is no vertical asymptote at x=0 for that function.

Slant Asymptotes

The provided sources do not contain any information regarding slant asymptotes.

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