Limits and Continuity in Calculus

Math 20A Study Notes

Instructor: Gaojin He
Department: Mathematics
Institution: University of California San Diego
Semester: Fall 2025

1. The Idea of Limit

1.1 Examples of Limits

Example 1: Finding Instantaneous Velocity

• A ball is dropped from the air at time t = 0.

• The displacement function is given by:
  $s(t) = 5t^2 ext{ meters}$

• To find the instantaneous velocity at t = 1s, we approximate this value by finding the average speed over the interval [1, 1 + ∆t] where ∆t is small.

• The average velocity is defined as:
  $v = rac{ ext{change in position}}{ ext{change in time}} = rac{∆s}{∆t} = rac{s(1 + ∆t) - s(1)}{∆t}$

• As ∆t → 0, we find that v → 10.

• Therefore, the instantaneous velocity at t = 1s is:
  $10 ext{ m/s}$

Example 2: Finding Tangent Lines

• We need to define the tangent line of the function
  $s(t) = 5t^2$
  at the specific point (1, 5).

Steps to Define the Tangent Line:

1. Point: The tangent line must pass through (1, 5).

2. Slope: The slope of this line should approximate the slope of the secant line connecting (1, 5) and (1 + ∆t, 5(1 + ∆t)^2) on the graph.

3. Calculation of Slope: The slope of the secant line is: $rac{5(1 + ∆t)^2 - 5(1)}{∆t} = rac{5[(1 + ∆t)^2 - 1]}{∆t} = 10 + 5∆t$

   • As ∆t → 0, the slope approaches 10.

4. Tangent Equation: Thus, the equation of the tangent line can be derived as:
   $y - 5 = 10(x - 1) <br>ightarrow y = 10x - 5$

1.2 Implications of Limits

• The instantaneous velocity at time t0 can be described as the slope of the tangent line at the point (t0, s(t0)).

• Fundamental limit principles demonstrate the behavior of functions as independent variables approach certain values.

2. Investigating Limits

2.1 Definition and Existence of Limits

• Consider the function
  $f(x) = rac{<br>in sin x}{x}$
  which is undefined at x = 0.

• Behavior Near Zero: The limit is investigated as follows:

  • If x approaches 0, then sin x also approaches 0, leading to finite ratios.

2.2 Formal Definition of Limits

• Let L be a finite number.

• If f(x) is defined for all x around c (open interval) but potentially not at c:

  • We declare
    $ext{lim}_{x o c} f(x) = L$
    or
    $f(x) o L ext{ as } x o c$
    if the value of |f(x) - L| can be made arbitrarily small as x approaches c.

• If L does not exist, we denote this by:
  $ext{lim}_{x o c} f(x) ext{ DNE}$.

2.3 Examples and Theorems on Limits

• Theorem: For linear functions
  $f(x) = ax + b$, the limit is defined as:
  $ext{lim}_{x o c} f(x) = f(c) = ac + b$.

• Example Function:
  $g(x) = 3x + 1$
  leads to:
  $ext{lim}_{x o 4} g(x) = 13$

3. The Concept of Continuity

3.1 Definitions of Continuity

• Left-Continuous at a point c if:
  $ext{lim}_{x o c^-} f(x) = f(c)$

• Right-Continuous at a point c if:
  $ext{lim}_{x o c^+} f(x) = f(c)$

• A function f is continuous at c if both limits exist and equal to the function value:
  $ext{lim}_{x o c} f(x) = f(c)$

3.2 Types of Discontinuities

• Removable Discontinuity: Exists when
  $ext{lim}<em>{x o c} f(x)$ exists but does not equal f(c) or f(c) is undefined. Example: $ext{Lim}</em>{x o 2} f(x) = 5 ext{ but } f(2) = 10$.

• Jump Discontinuity: Both one-sided limits exist but are not equal.

• Infinite Discontinuity: One or both one-sided limits approach infinity at c.

• Essential Discontinuity: Limits do not converge or approach infinity.

4. Differentiation

4.1 Definition of Derivative

• The derivative of a function f at a point a is defined as:
  $f'(a) = ext{lim}_{h o 0} rac{f(a + h) - f(a)}{h}$

4.2 Tangent Line and Application

• The tangent line's equation at point (a, f(a)) can be constructed as:
  $y - f(a) = f'(a)(x - a)$

4.3 Higher-Order Derivatives

• n-th Order Derivative (denoted as f(n)(x)): Function obtained by deriving n times.

• Example: For s(t) = 25t - 0.3t^3:

  • First derivative:
    $s'(t) = 25 - 0.9t^2$

  • Second derivative:
    $s''(t) = -1.8$

• Acceleration interpreted as the second derivative.

4.4 Rates of Change

• The derivative represents the instantaneous velocity when applied to displacement over time.

• Example: Area function
  $A(r) = ext{π}r^2$ has a derivative
  $A'(r) = 2 ext{π}r$ representing the circumference of a disk.

5. Trigonometric Functions

5.1 Definitions and the Unit Circle

• Trigonometric functions defined as ratios in a right triangle.

• Usage of radian as the preferred measure:
  $heta = rac{s}{r}$ where s is arc length on the unit circle.

5.2 Fundamental Theorems about Derivatives

• Theorems regarding primary trigonometric functions:
  $rac{d}{dx}( ext{sin } x) = ext{cos } x$
  $rac{d}{dx}( ext{cos } x) = - ext{sin } x$.

• Lemmas (Addition Formulas):
  $ext{sin}( heta imes ext{cos}(\beta) + ext{sin}(\beta))</p></li></ul><h4 id="bf8d781c-d648-4331-bf01-89d478a325f1" data-toc-id="bf8d781c-d648-4331-bf01-89d478a325f1" collapsed="false" seolevelmigrated="true">6. The Chain Rule</h4><h5 id="98791ae9-44c2-4316-acb7-2ab8a7c1a7eb" data-toc-id="98791ae9-44c2-4316-acb7-2ab8a7c1a7eb" collapsed="false" seolevelmigrated="true">6.1 Definition and Applications</h5><ul><li><p>Provided if <strong>f</strong> and <strong>g</strong> are differentiable, the chain rule states: <br>$ rac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$</p></li></ul><h5 id="75668337-6bf2-47ff-a3b1-82b17771de9b" data-toc-id="75668337-6bf2-47ff-a3b1-82b17771de9b" collapsed="false" seolevelmigrated="true">6.2 Example of Applying the Chain Rule</h5><ul><li><p>To compute <br>$ rac{d}{dx}[ ext{sin}(2x)] = 2 ext{cos}(2x)$.</p></li><li><p>For double compositions, apply the chain rule repetitively.</p></li></ul><h4 id="6fe02678-b0f0-4c26-ab85-75e7b689e659" data-toc-id="6fe02678-b0f0-4c26-ab85-75e7b689e659" collapsed="false" seolevelmigrated="true">7. Implicit Differentiation</h4><h5 id="39aaa69f-fbb9-4acd-bc0b-3d36b64757e3" data-toc-id="39aaa69f-fbb9-4acd-bc0b-3d36b64757e3" collapsed="false" seolevelmigrated="true">7.1 Concept Overview</h5><ul><li><p>Tangent lines can exist even if the function is not explicitly given.</p></li></ul><h5 id="5a6f0fc4-d101-43f2-af7c-059002f65649" data-toc-id="5a6f0fc4-d101-43f2-af7c-059002f65649" collapsed="false" seolevelmigrated="true">7.2 Example: Implicit Differentiation</h5><ul><li><p>Given the circle equation <strong>x² + y² = 1</strong>, finding tangent involves differentiating both sides implicitly: <br>$2x + 2y rac{dy}{dx} = 0
  ightarrow rac{dy}{dx} = - rac{x}{y}$$.