Calculus Study Notes: Limits, Trigonometric Identities, and the Unit Circle

Limits and Domains in Calculus

Domain of a Function

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. Determining the domain is essential before evaluating limits or performing calculus operations.

• Rational Functions: For $f(x) = \frac{1}{g(x)}$, the domain excludes values where $g(x) = 0$ (since division by zero is undefined).

• Even and Odd Functions:

  • If $y = f(x)$ is odd, the domain is all real numbers $x \in \mathbb{R}$ (unless the denominator is zero).

  • If $y = f(x)$ is even, the domain is $x \geq 0$ or $x \leq 0$ depending on the context.

• Solving for Domain: To find the domain, solve for all $x$ where the denominator is not zero and any other restrictions (such as square roots requiring non-negative radicands).

Limits of Sine and Cosine Functions

Key Limit Formulas as x Approaches 0

These fundamental limits are used frequently in calculus, especially when evaluating limits involving trigonometric functions.

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• $\lim_{x \to 0} \frac{\tan x}{x} = 1$

• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

Trigonometric Identities for Simplifying Limits

Pythagorean Identities

Pythagorean identities relate the squares of sine, cosine, and tangent functions. They are useful for simplifying expressions before taking limits.

• $\sin^2(x) + \cos^2(x) = 1$

• $1 + \tan^2(x) = \sec^2(x)$

• $1 + \cot^2(x) = \csc^2(x)$

Reciprocal Identities

Reciprocal identities express trigonometric functions in terms of their reciprocals.

• $\sin(x) = \frac{1}{\csc(x)}$

• $\cos(x) = \frac{1}{\sec(x)}$

• $\tan(x) = \frac{1}{\cot(x)}$

Simplifying Trigonometric Expressions

• To simplify $\frac{\tan(x)}{\sin(x)}$, use $\tan(x) = \frac{\sin(x)}{\cos(x)}$:

• $\frac{\tan(x)}{\sin(x)} = \frac{\sin(x)/\cos(x)}{\sin(x)} = \frac{1}{\cos(x)}$

• To simplify $\frac{\sin(x)}{\tan(x)}$, use $\tan(x) = \frac{\sin(x)}{\cos(x)}$:

• $\frac{\sin(x)}{\tan(x)} = \frac{\sin(x)}{\sin(x)/\cos(x)} = \cos(x)$

Double Angle Identities

Double angle identities are useful for rewriting trigonometric expressions involving $2x$.

• $\sin(2x) = 2\sin(x)\cos(x)$

• $\cos(2x) = \cos^2(x) - \sin^2(x)$

Unit Circle

The unit circle is a circle of radius 1 centered at the origin. It is fundamental in trigonometry for understanding the values of sine and cosine at various angles.

• Each point on the unit circle corresponds to $(\cos(\theta), \sin(\theta))$ for an angle $\theta$.

• Common angles (in degrees and radians) and their coordinates are shown on the unit circle diagram.

• For example, at $\theta = 0$ ($0^\circ$), the coordinates are $(1, 0)$; at $\theta = \frac{\pi}{2}$ ($90^\circ$), the coordinates are $(0, 1)$.

Steps to Prove a Limit Exists

Formal Definition of a Limit

To prove $\lim_{x \to c} f(x) = L$:

1. Step 1: Assume $\lim_{x \to c} f(x) = L$ exists.

2. Step 2: Show that for every $\varepsilon > 0$, there exists $\delta > 0$ such that whenever $0 < |x - c| < \delta$, $|f(x) - L| < \varepsilon$.

This is the rigorous (epsilon-delta) definition of a limit in calculus.

Summary Table: Trigonometric Identities

Identity Type	Formula
Pythagorean	$\sin^2(x) + \cos^2(x) = 1$
Pythagorean	$1 + \tan^2(x) = \sec^2(x)$
Pythagorean	$1 + \cot^2(x) = \csc^2(x)$
Reciprocal	$\sin(x) = \frac{1}{\csc(x)}$
Reciprocal	$\cos(x) = \frac{1}{\sec(x)}$
Reciprocal	$\tan(x) = \frac{1}{\cot(x)}$
Double Angle	$\sin(2x) = 2\sin(x)\cos(x)$
Double Angle	$\cos(2x) = \cos^2(x) - \sin^2(x)$

Example Application: To evaluate $\lim_{x \to 0} \frac{\sin(2x)}{x}$, use the double angle identity: $\sin(2x) = 2\sin(x)\cos(x)$, so $\frac{\sin(2x)}{x} = \frac{2\sin(x)\cos(x)}{x}$. As $x \to 0$, $\sin(x)/x \to 1$ and $\cos(x) \to 1$, so the limit is 2.

Additional info: The notes above provide foundational concepts for calculus students, especially for evaluating limits involving trigonometric functions and understanding the rigorous definition of a limit.