W6_Limits and continuity

Fundamentals of Mathematics 1 Session 6: Limits and Continuity

Instructor: Erwan LamySchool: ESCP BUSINESS SCHOOL, ParisAffiliated with IT ALL STARTS HERE BERLIN & WARSAW

Objectives

• Understand limits and their basic properties.

• Understand one-sided limits, infinite limits, and limits at infinity.

• Understand continuity and identify points of discontinuity for a function.

• Solve nonlinear inequalities.

Outlines

• Limits of Function (Chapter 10.1 & 10.2)

• Continuity (Chapter 10.3)

• Continuity Applied to Inequalities (Chapter 10.4)

Limits of Function (Chapters 10.1 & 10.2)

Definition of Limit

Let f(x) be a real-valued function and c a real number. The limit of f(x) as x approaches c is the value L that f(x) approaches as x gets close to c.Notation: lim_{x→c} f(x) = LImportant: x never reaches c and f(x) never reaches L. If f(x) is continuous at x=c: lim_{x→c} f(x) = f(c).

Methods to Calculate Limits

• Direct Substitution: Substitute the value of c directly into the function. If f(c) is defined, then lim_{x→c} f(x) = f(c).

• Factoring: Factor the function when direct substitution results in an indeterminate form (like 0/0). Simplify and then apply direct substitution again.

• Rationalization: For functions involving square roots, multiply the numerator and denominator by the conjugate to eliminate the root.

• Limit Laws: Apply limit properties (e.g., linearity, multiplication) to break down complex functions into simpler parts.

Example: Zeno's Paradox

Achilles races to a tree: He must get halfway there, then a quarter of the way, and so on. He approaches the tree indefinitely without ever reaching it. The limit of his running is the position of the tree.

Practical Examples of Limits

Example: lim_{x→4} √x = 2 as x approaches 4.Additional Limit Examples

• f(x) not defined at x=0: lim_{x→0} f(x) = 0

• Piecewise functions: f(x) = {2 for x < 1, 2 for x = 1, 2 for x > 1} lim_{x→1} f(x) = 1 ≠ f(1) = 2

One-Sided Limits

Limits can approach from the left or the right:

• Left limit: lim_{x→c-} f(x)

• Right limit: lim_{x→c+} f(x)

Methods for One-Sided Limits

• Evaluate: Determine the limit by substituting values that approach c from the left (for left limit) and from the right (for right limit).

• Check Equality: Determine if both one-sided limits are equal to confirm existence: lim_{x→c} exists if lim_{x→c-} f(x) = lim_{x→c+} f(x).

Examples of One-Sided Limits

• f(x) not defined at x=0: lim_{x→0-} f(x) = 0, lim_{x→0+} f(x) = 0

• Discontinuity example: lim_{x→0-} f(x) = -1, lim_{x→0+} f(x) = 1 (both don't exist).

Infinite Limits

Definition: lim_{x→a} f(x) = ∞ if f(x) becomes arbitrarily large. lim_{x→a} f(x) = -∞ if f(x) becomes arbitrarily large and negative.

Methods for Infinite Limits

• Analyze behavior: Evaluate f(x) as x approaches a to see if it tends towards positive or negative infinity.

• Identify vertical asymptotes: Determine where the function does not exist to evaluate limits at points of discontinuity.

Example of Infinite Limits

• f(x) = 1/x: lim_{x→0-} = ∞, lim_{x→0+} = ∞

Limits at Infinity

Definition: lim_{x→∞} f(x) = L if f(x) approaches L as x becomes infinitely large.Notation: lim_{x→-∞} f(x) = L for large negative x.

Methods for Evaluating Limits at Infinity

• Horizontal asymptotes: Analyze the degrees of the polynomial in the numerator and denominator for rational functions.

• Dominant terms: Identify the term with the highest degree in the numerator and denominator to estimate the limit.

Example of Limits at Infinity

• Example: lim f(x) as x approaches constants or in rational functions: Observe behavior of the numerator and denominator.

Properties of Limits

• Constant Function: lim_{x→c} k = k

• Linearity: lim_{x→c}[af(x) + b] = a lim_{x→c}f(x) + b

• Multiplication: lim_{x→c}[f(x)g(x)] = lim_{x→c}f(x) * lim_{x→c}g(x)

Continuity (Chapter 10.3)

Definition of Continuity

A function f is continuous at a point c if: f(c) = lim_{x→c} f(x)

Methods to Identify Continuity

• Evaluate limits from both sides and compare to f(c).

• Check domain: Ensure that f(c) is defined for continuity.

Continuous functions include:

• All polynomial functions.

• Rational functions, except where denominator = 0.

Examples of Continuous and Discontinuous Functions

Example function: f(x) = {2 for x < 1, 2 for x = 1, 2 for x > 1} (discontinuous at x=1)Discontinuity example: f(x) = {1 for x ≥ 0, -1 for x < 0}, discontinuous at x=0.

Infinite Discontinuity

Definition: A function has infinite discontinuity at a point a if any one-sided limit approaches ±∞.

Example of Infinite Discontinuity

• f(x) = 1/(x-1) has infinite discontinuity at x=1.