Homework Assignments and Quizzes

• The upcoming quiz is based on the previous week's homework.
• This week's homework focuses on sections 2.6 and 2.7, as decided by the instructors.
• With less homework this week, extra problems will be provided for review of section 2.5.

Review of Previous Concepts

• In section 2.5, algebraic manipulation was applied to analyze polynomials and rational functions. This included identifying zeros, infinity limits, and holes in the functions to sketch their behavior.

• Certain applications of algebraic manipulations were discussed, especially concerning trigonometric functions, which differ fundamentally from polynomial or rational functions.

• Sine and cosine functions are bounded within the interval [-1, 1]. A graphical representation illustrates that:

  • For any input x, the outputs of both sine and cosine will always lie between -1 and 1.
  • Notably, the function ( \sin \left( \frac{1}{x} \right) ) approaches infinity as x approaches zero, but remains bounded.

Introduction of Trigonometric Concepts

• Using algebraic manipulation is limited with trigonometric functions compared to polynomials. Thus, the session will focus on understanding the properties of sine and cosine.

• Further assertions note:

  • ( \sin(x) ) is always within the range of [-1, 1].
  • Similarly, ( \cos(x) ) is confined within the same bounds.

• Implications of these bounds for functions approaching limits, emphasizing:

  • The behavior of ( \sin \left( \frac{1}{x} \right) ) regardless of how x approaches positivity or negativity will always yield outputs between -1 and 1.

The Squeeze Theorem

• The central theorem for today's discussion is the Squeeze Theorem, denoted as THM. The theorem provides a method for finding the limit of a function ( f ) at a point ( c \
  when upper and lower bounds are established.

• Specific statement of the theorem:

  • If for all ( x \neq c ) in some interval around c, the following holds:
    • ( l(x) \leq f(x) \leq u(x) )
  • Where ( l ) and ( u ) represent functions providing the lower and upper bounds, respectively.

• Visualization of bounds:

  • Essentially, this scenario places ( f ) in between the functions ( l ) and ( u ), indicating the limit structure surrounding ( c ).

• Two essential conditions for the application of the theorem:

  • ( \lim{x \to c} l(x) = \lim{x \to c} u(x) = l ), indicating that both bounds have the same limit approaching c.
  • This implies that the limit of the function ( f ) will, therefore, also equal ( l ).

• Therefore, if a function is bounded by two others that converge to the same limit, the squeezed function must also converge to that limit. This establishes the existence of the limit.

Example Application of the Squeeze Theorem

• Consider ( f(x) = x \sin \left( \frac{1}{x} \right) ).

• Assessment of upper and lower bounds yields:

  • Since ( \sin(x) ) is constrained within [-1, 1], we uncover that:
    -( f(x) \leq x ) and ( f(x) \geq -x ).
  • The graph depicts that ( f ) oscillates between these two linear bounds as x approaches 0.

• Calculation of limits approach:

  • To find ( \lim_{x \to 0} f(x) ):
    • Following the bounds yields:
    • ( \lim{x \to 0} x = 0 ) and ( \lim{x \to 0} -x = 0 ).
    • Therefore, ( \lim_{x \to 0} f(x) = 0 ).

Important Trigonometric Limits

• Theorem stating:

  • ( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 )
  • ( \lim_{\theta \to 0} (1 - \cos \theta) = 0 )

• The proof contextually utilizes the Squeeze Theorem:

  • To establish relationships like ( \cos \theta \leq \frac{\sin \theta}{\theta} \leq 1 ) for discourse in this body.

• Statements confirm that continuity laws dictate that both ( \cos \theta ) maintains a limit of 1 as ( \theta \to 0 ).

Proving Limits Using Theorem Two

• The example shows how limits can derive from theorem applications in different setups, emphasizing on connections between sine and cosine functions.
• Establishing continuity: If ( g ) is defined appropriately through transformations to maintain a continuous function, theorems from previous chapters affirm consistent application to reach limits.

Practical Examples and Applications

• Providing practical applications for limit determinations, includes challenges, workouts, and tricky instances to build fluency in finding and manipulating limits accurately.
• Engaging with key functions, honing in on trigonometric limits through innovative problem-solving approaches, and leveraging theorem connections to broaden understanding and application.