Calculus: Limits and Function Behavior

Introduction to the Concept of Slope

• The discussion begins with understanding how to find the slope between two points, referred to as point $p$ and point $q$.

• The formula for slope between two points is:
  $m_{pq} = \frac{f(q) - f(p)}{q - p}$

• This formula computes the average slope over the interval defined by units $p$ and $q$.

• The average slope provides an approximation of the slope at point $p$.

• As the distance, denoted as $\Delta x$, between the points $p$ and $q$ is reduced, this average slope increasingly approximates the instantaneous slope at point $p$.

• Instantaneous slope corresponds to the slope of the tangent line to the curve at point $p$.

The Limit Process in Slope Calculation

• To find the instantaneous slope directly at point $p$, we cannot simply set $\Delta x$ to zero due to the undefined nature of the slope in that scenario.

• Instead, the limit is applied:
  $m<em>{pq} = \lim</em>{q \to p} m_{pq}$

• Therefore, we find:
  $\lim_{q \to p} \frac{f(q) - f(p)}{q - p}$

• This approach defines how we can compute the slope of the tangent line at point $p$ by letting $\Delta x$ approach zero, without actually reaching zero.

Homework Access and Grades

• The speaker addresses the technicalities concerning student access to homework assignments on their platform and the potential discrepancies in grades listed on Canvas versus WebAssign.

• It is emphasized:

  • The accurate reflection of students' performance is found in WebAssign only.

Function Analysis Example: $m(x) = \frac{x^3 - 1}{x - 1}$

• A particular function, $m(x) = \frac{x^3 - 1}{x - 1}$, is discussed to illustrate limit concepts.

• When $x = 1$, plugging in results in the indeterminate form:
  $\frac{0}{0}$

• This creates an indeterminate form since the denominator becomes zero, rendering the expression undefined at that specific input.

Understanding Functions via Limit Approximation

• The notion of limits allows us to evaluate the function's behavior near point $x = 1$ by substituting values increasingly close to 1:

  • For $x = 0$: Output is 1.

  • For $x = -1$: Output is 1.

  • For $x = -2$: Output is 3.

• A pattern emerges suggesting the function might resemble a parabola, which is verified by testing more values around the area of interest.

Factorization and Limits in Function Behavior

• As we substitute different values (e.g., $x = 2, 1.1, 1.01$), we evaluate how close outputs are to a specific value.

• Factorization of the cubic expression helps explore the function more deeply:

  • The cubic polynomial factors into:
    $x^3 - 1 = (x - 1)(x^2 + x + 1)$

  • Thus, the function can be simplified as:
    $m(x) = x^2 + x + 1$

• This simplification highlights that a graphical behavior exists with a removable discontinuity at $x = 1$, where the function is otherwise undefined.

Exploring Other Limits and Their Behaviors

• Several cases are discussed regarding limits that do not exist:

  • Case 1: Directional Disagreement - The limit does not approach the same value from either side, denoted as: $\lim_{x \to 0} \frac{|x|}{x}$.

  • Case 2: Infinite Growth - The function approaches infinity, such as: $\lim_{x \to 0} \frac{1}{x^2}$.

  • Case 3: Infinite Oscillation - The function (e.g., sine) oscillates infinitely without stability: $\lim_{x \to 0} \sin\left(\frac{1}{x}\right)$.

Formal Definition of Limit

• A rigorous definition is proposed: for a function $f$ defined on a neighborhood of $c$ (except possibly at $c$ itself), the limit is $L$ if:

  • For every \epsilon > 0, there exists a \delta > 0 such that:
    0 < |x - c| < \delta \implies |f(x) - L| < \epsilon

Conclusion and Forward Look

• The lecture concludes with a look ahead to upcoming classes, including pretests and engaging with additional examples of limits.

• Students should be prepared with their laptops for assessments and further exploration of limits and continuity in functions.