Introduction to Tangents and Derivatives Using Limits

Lecture Learning Outcomes

• Find the equation of the tangent to a curve at a point

• Find the derivative of a function using the limit definition

Properties of Continuous Functions

Definition and Theorems

A function is continuous at a point if its value and its limit at that point are equal. Several properties and types of functions are known to be continuous within their domains.

• If f and g are continuous at a and c is a constant, then:

  • f + g is continuous at a

  • f - g is continuous at a

  • f × g is continuous at a

  • f / g is continuous at a, provided $g(a) \neq 0$

  • c × f is continuous at a

Types of Continuous Functions

The following types of functions are continuous everywhere in their domains:

Type	Domain
Polynomials	$(-\infty, \infty)$
Rational functions	$Q(x) \neq 0$
Root functions	$n$ odd: $(-\infty, \infty)$ $n$ even: $[0, \infty)$
Trigonometric functions	$(-\infty, \infty)$ (with exceptions for some functions)
Inverse trigonometric functions	Domain-specific
Exponential functions	$(-\infty, \infty)$
Logarithmic functions	$(0, \infty)$

• Example: $\sin x$ and $\cos x$ are continuous for all $x \in (-\infty, \infty)$.

Equations of Lines and Tangents

Equation of a Line

The general equation of a line passing through point $(x_1, y_1)$ with slope $m$ is:

• $y - y_1 = m(x - x_1)$

• Slope formula: $m = \frac{\Delta y}{\Delta x}$

Tangent to a Curve

To find the tangent line to a curve $y = f(x)$ at a point $P(a, f(a))$, consider a nearby point $Q(x, f(x))$ and compute the slope of the secant line $PQ$:

• $m_{PQ} = \frac{f(x) - f(a)}{x - a}$

As $Q$ approaches $P$ along the curve (i.e., as $x \to a$), the secant slope approaches the tangent slope:

• $m_{\text{tangent}} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

The equation of the tangent line at $x = a$ is:

• $y - f(a) = m_{\text{tangent}} (x - a)$

Alternative Tangent Slope Formula

Another formula for the slope of the tangent line at $P(a, f(a))$ is:

• $m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Examples

• Example 1: Find the equation of the tangent to $y = \sqrt{1 - 3x}$ at $(-1, 2)$ Solution: $m = \lim_{x \to -1} \frac{\sqrt{1 - 3x} - 2}{x + 1}$ After simplification, $m = -\frac{3}{4}$ Equation: $y - 2 = -\frac{3}{4}(x + 1)$

• Example 2: Find the equation of the tangent to $y = \frac{3}{x}$ at $(3, 1)$ Solution: $m = \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h}$ $m = \lim_{h \to 0} \frac{\frac{3}{3 + h} - 1}{h}$ After simplification, $m = -\frac{1}{3}$ Equation: $y - 1 = -\frac{1}{3}(x - 3)$

Derivatives and the Limit Definition

Definition of the Derivative

The derivative of a function $f$ at a number $a$, denoted $f'(a)$, is defined as:

• $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• Alternatively: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Example: Derivative Using the Limit Definition

• Example: Find $f'(3)$ for $f(x) = \frac{x^2}{x + 6}$ Solution: $f'(3) = \lim_{x \to 3} \frac{f(x) - f(3)}{x - 3}$ $= \lim_{x \to 3} \frac{\frac{x^2}{x + 6} - 1}{x - 3}$ $= \lim_{x \to 3} \frac{x^2 - (x + 6)}{(x + 6)(x - 3)}$ $= \lim_{x \to 3} \frac{(x - 3)(x + 2)}{(x + 6)(x - 3)}$ $= \lim_{x \to 3} \frac{x + 2}{x + 6}$ $= \frac{3 + 2}{3 + 6} = \frac{5}{9}$

Derivative Function

The derivative function $f'(x)$ gives the derivative at any point $x$:

• $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Examples: Finding Derivative Formulas

• Example a: $f(x) = x^3 - x$ $f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3 - h}{h}$ Expand $(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$ $f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h}$ $= \lim_{h \to 0} 3x^2 + 3xh + h^2 - 1$ $= 3x^2 - 1$

• Example b: $g(x) = \sqrt{x - 1}$ $g'(x) = \lim_{h \to 0} \frac{\sqrt{x + h - 1} - \sqrt{x - 1}}{h}$ Multiply numerator and denominator by the conjugate: $= \lim_{h \to 0} \frac{1}{\sqrt{x + h - 1} + \sqrt{x - 1}}$ $= \frac{1}{2\sqrt{x - 1}}$

Summary Table: Limit Definitions

Concept	Limit Definition
Slope of tangent line at $x = a$	$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$
Derivative at $x = a$	$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
Derivative function	$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Key Terms

• Continuous function: A function with no breaks, jumps, or holes in its domain.

• Tangent line: A straight line that touches a curve at a single point and has the same slope as the curve at that point.

• Derivative: The instantaneous rate of change of a function at a point; the slope of the tangent line.

• Limit: The value that a function approaches as the input approaches a certain point.

Applications

• Finding the slope of a curve at a point (instantaneous rate of change)

• Determining the equation of the tangent line to a curve

• Calculating derivatives using the limit definition for various functions

Additional info: The notes also include step-by-step worked examples for finding tangent lines and derivatives using both the $x \to a$ and $h \to 0$ limit definitions, which are foundational for understanding calculus concepts such as differentiation and instantaneous rates of change.