Calculus 1 Lecture 4: Limits

Calculus 1 Lecture 4: Chapter 2 (2.2)

• Definition of Limit:
  • A function f is defined for all x near a, except possibly at a. If f(x) is arbitrarily close to L (as close to L as we like) for all x sufficiently close to a, but not equal to a, we write lim f(x) = L. This means the limit of f(x) as x approaches a equals L.

Calculus 1 Lecture 4: Examlpe 1

• Finding Limits from a Graph:
  • The graph of a function can be used to determine specific function values (f(x)) and the limit of function as x approaces a certain value (lim f(x)).

Calculus 1 Lecture 4: Solution (Example 1, a)

• f(1) and lim f(x) as x approaches 1:
  • f(1) = 2
  • lim f(x) as x approaches 1 is 2.

Calculus 1 Lecture 4: Solution (Example 1 b)

• f(2) and lim f(x) as x approaches 2:
  • f(2) = 5
  • lim f(x) as x approaches 2 is 3

Calculus 1 Lecture 4: Solution (Example 1 c)

• f(3) and lim f(x) as x approaches 3:
  • f(3) is undefined.
  • lim f(x) as x approaches 3 is 4.

Calculus 1 Lecture 4: Another Example

• Finding Limits from a Table:
  • A table of values of f(x) corresponding to values of x near 1 can help to determine the limit of the function f(x) as x approaches 1 to be 0.5

Calculus 1 Lecture 4: One-Sided Limits

• Definition of One-sided limits:
  • Right-sided limit: lim f(x) = L means f(x) is arbitrarily close to L as x approaches a from the right (x > a).
  • Left-sided limit: lim f(x) = L means f(x) is arbitrarily close to L as x approaches a from the left (x < a).

Calculus 1 Lecture 4: Example 3

• Examining Limits Graphically and Numerically
  • The lecture uses tables and graphs of functions to explore concept of limits of functions. The function examples were (x³-8)/(4(x-2)).
  • Analysis of limit of functions as x approaches 2

Calculus 1 Lecture 4: Solution (Example 3)

• Right-sided limit
  • Shows f(x) approaches 3 as x becomes arbitrarily close to 2 on the right hand side.
• Left-sided limit
  • Shows f(x) approaches 3 as x becomes arbitrarily close to 2 on the left hand side.
• Two-sided limit
  • The limit of f(x) is 3 as x approaches 2

Calculus 1 Lecture 4: One-sided and Two-Sided Limits

• Relationship
  • A two-sided limit exists only if the left-sided limit and right-sided limit are equal to each other. Otherwise the limit does not exist.

Calculus 1 Lecture 4: Example 4

• Function with a Jump:
  • Demonstrates finding limits associated with a discontinuous function that has an open/closed circle

Calculus 1 Lecture 4 (2.3): Techniques for Computing Limits

• Limits of Linear Functions
  • For any linear function f(x) = mx + b, lim f(x) = f(a).

Calculus 1 Lecture 4: Example 1 (Linear Functions)

• Limits of Linear Functions
  • The examples in this lecture illustrate finding limits of linear functions, providing step-by-step computations.

Calculus 1 Lecture 4: Limit Laws

• Theorem 2.3:
  • Outlines Limit Laws that provide rules needed to compute various limits.

Calculus 1 Lecture 4 Example 2

• Evaluating Limits:
  • Show various situations where limit laws are applicable including simple polynomials, and functions that have to be modified (rationalized etc.)

Calculus 1 Lecture 4: Example 2 (Continued)

• Limit Laws (Example 2 Continued):
  • Demonstrates how the limit laws are used to compute different types of limits of functions.

Calculus 1 Lecture 4: Limits of Polynomial and Rational Functions

• Theorem 2.4:
  • Shows that for polynomial and rational functions, the limit at a point a is simply the result of substituting x= a into the function. This is true provided the denominator does not become zero

Calculus 1 Lecture 4: One-Sided Limits

• Theorem 2.3 (Continued):
  • Details the implications and conditions for both one-sided and two-sided limits (n>0 integer)

Calculus 1 Lecture 4: Example 5

• Calculating Left- and Right-sided Limits:
  • Demonstrates an example on how to compute the one sided limits of a piecewise defined function

Calculus 1 Lecture 4: Examples Using Other Techniques

• Other Techniques: Limit Evaluations
  • Examines different techniques to compute limits including the conjugate method, and factoring the numerator/denominator.

Calculus 1 Lecture 4: Squeeze Theorem

• Theorem 2.5: The Squeeze Theorem:
  • Provides a theorem to evaluate a limit by applying the limit laws in a specific scenario with an example (x²sin(1/x) as x→ 0).

Calculus 1 Lecture 4: Infinite Limits (2.4)

• Definition of Infinite Limits:
• Outlines properties for handling infinite limits of functions

Calculus 1 Lecture 4: Infinite Limits (Table 2.6)

• Table 2.6 (Infinite Limits):
  • Analyzes the function f(x)=1/x² to find lim f(x) as x→0. Gives more detailed information on the infinite limiting cases

Calculus 1 Lecture 4: Definition (Infinite Limits) and (One-Sided Infinite Limits)

• Definition:
  • Describes the conditions for infinite limits including one sided infinite limits of functions.

Calculus 1 Lecture 4: Vertical Asymptote

• Vertical Asymptote:
  • Defines the conditions needed to identify a vertical asymptote in the analysis of functions.

Calculus 1 Lecture 4: Example 2 (Vertical Asymptotes)

• Limits Graphically:
  • Explores finding limits at vertical asymptotes of function using graphs using the concept of one-sided limits.

Calculus 1 Lecture 4: Determining Limits Analytically (Example 3)

• Analytically determining limits
  • Shows how to determine limits of functions using algebra rather than just graphs, including using factoring and algebraic manipulation to simplify the function to compute the limit.

Calculus 1 Lecture 4: Example 2 (Analytic Approach)

• Analytic Approach to Limits:
• Shows another example on finding limits analytically as opposed to graphically.

Calculus 1 Lecture 4: Example 3: Location of Vertical Asymptote

• Location of Vertical Asymptotes:
  • Examines finding vertical asymptotes in function based on finding applicable limit using step by step algebraic and graphical methods.

Calculus 1 Lecture 4: Example 4

• Limits of Trigonometric Functions:
  • Demonstrates how limit laws could be extended to deal with trigonometric function using graphical examples

This quiz focuses on the definition of limits in calculus, particularly as discussed in Chapter 2 (2.2). It includes examples of finding limits from a graph and provides specific function values as limits are approached. Test your understanding of these fundamental concepts in calculus!