Calculus Continuity Overview

Continuity

• A function f is continuous at a point a if the following three conditions are met:
• f(a) is defined (i.e., a is in the domain of f).
• lim_x→a f(x) exists.
• lim_x→a f(x) = f(a).
• A function is continuous on an interval if it is continuous at every point in the interval.
• Discontinuities:
• Removable discontinuity: A discontinuity that can be "removed" by redefining the function at a single point. This occurs when lim_x→a f(x) exists but is not equal to f(a).
• Jump discontinuity: A discontinuity where the limit from the left and the limit from the right exist but are not equal.
• Infinite discontinuity: A discontinuity where the function approaches infinity (positive or negative) as x approaches a particular value.
• Intermediate Value Theorem: If f is continuous on the closed interval [a, b] and k is any value between f(a) and f(b), then there exists a value c in the interval (a, b) such that f(c) = k.

Limits

• A limit describes the value that a function approaches as the input (x) approaches a certain value.
• Calculating limits:
• Direct substitution: If the function is continuous at the point, substitute the value directly into the function to find the limit.
• Factorization and cancellation: Cancel common factors in the numerator and denominator to simplify the expression before evaluation.
• Rationalizing techniques: Use conjugates to eliminate radicals in the expression.
• L'Hôpital's rule: Use derivatives to evaluate limits in indeterminate forms (e.g., ∞/∞, 0/0).
• Types of limits:
• One-sided limits (e.g., lim_x→a^- f(x) or lim_x→a⁺ f(x)) indicate the limit as x approaches a from the left or right, respectively.
• Limit properties: Rules of limits allow us to find limits of complex functions by breaking them down into simpler functions.
• Infinite limits: The limit is infinity if the function grows without bound as x gets close to a certain value. As x approaches a limit, the values get arbitrarily large or arbitrarily small.

Differentiability

• A function f is differentiable at a point a if the limit defining its derivative exists at that point.
• The derivative of a function f at a point a, denoted as f′(a), represents the instantaneous rate of change of f at a. This describes the slope of the tangent line to the curve f at a.
• The derivative is a function that gives the slope of the tangent line at every point on the graph of the original function.
• Relationship between differentiability and continuity:
• If a function is differentiable at a point, it must be continuous at that point.
• The converse is not necessarily true. A continuous function may not be differentiable at a point (e.g., a sharp corner or a vertical tangent).

Asymptotes

• Asymptotes are lines that a curve approaches as x or y approaches infinity.
• Vertical Asymptotes:
• Occur when the denominator of a rational function equals zero and the numerator is non-zero. Vertical asymptotes are found where the function approaches positive or negative infinity.
• Horizontal Asymptotes:
• Occur when the function approaches a constant value as x approaches infinity.
• Determine the behavior of the function as the input values become very large or very small.
• Case 1: When the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the horizontal asymptote is y = 0.
• Case 2: When the degrees are equal, the horizontal asymptote is y = the ratio of the leading coefficients.
• Case 3: When the numerator degree is higher than the denominator degree, there is no horizontal asymptote. The function will increase or decrease without bound.

This quiz covers the concept of continuity in functions, detailing the conditions for a function to be considered continuous at a point. It also explores various types of discontinuities, such as removable, jump, and infinite discontinuities, along with the Intermediate Value Theorem. Prepare to test your understanding of these critical calculus concepts.