Calculus Study Notes: Limits and Derivatives (Exam Practice)

Limits and Derivatives: Key Concepts and Practice

Limits: Definitions and Properties

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach a particular value. Understanding limits is essential for studying continuity, derivatives, and integrals.

• Precise Definition of Limit: The formal (epsilon-delta) definition states that for a function f(x), $\\lim_{x \to a} f(x) = L$ means: For every $\\varepsilon > 0$, there exists $\\delta > 0$ such that $|f(x) - L| < \\varepsilon$ whenever $0 < |x - a| < \\delta$.

• One-Sided Limits: Limits can be taken from the left ($x \to a^-$) or right ($x \to a^+$).

• Non-Existence of Limits: A limit may not exist if the function approaches different values from the left and right, or if it grows without bound.

Example: Precise Definition of Limit

• Problem: Find $\\delta$ such that $|\\sqrt{x^2 + 7} - 3| < 0.5$ whenever $|x - 2| < \\delta$.

• Solution Outline:

  • Set $|\\sqrt{x^2 + 7} - 3| < 0.5$ and solve for $|x - 2|$.

  • Find the appropriate $\\delta$ value.

Limit Calculations and Proofs

Calculating limits often involves algebraic manipulation, recognizing indeterminate forms, and applying limit laws.

• Common Limit Laws:

  • $\\lim_{x \to a} [f(x) + g(x)] = \\lim_{x \to a} f(x) + \\lim_{x \to a} g(x)$

  • $\\lim_{x \to a} [f(x)g(x)] = \\lim_{x \to a} f(x) \\times \\lim_{x \to a} g(x)$

  • $\\lim_{x \to a} [cf(x)] = c \\lim_{x \to a} f(x)$ for constant $c$

• Special Trigonometric Limits:

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

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

• Indeterminate Forms: Limits such as $\\frac{0}{0}$ or $\\frac{\\infty}{\\infty}$ require algebraic techniques or L'Hospital's Rule.

Example: Proving a Limit

• Problem: Prove $\\lim_{x \to 0} \\frac{\\sin x}{x} = 1$.

• Solution Outline:

  • Use the squeeze theorem or Taylor expansion for $\\sin x$ near $x = 0$.

Evaluating Limits

Limits can be evaluated using substitution, factoring, rationalization, or recognizing standard forms.

• Example: $\\lim_{x \to 1} \\frac{x^2 - 1}{x - 1}$

  • Factor numerator: $x^2 - 1 = (x - 1)(x + 1)$

  • Cancel $(x - 1)$, substitute $x = 1$ to get $2$.

• Example: $\\lim_{x \to 0} \\frac{\\sin(2x)}{x}$

  • Use $\\lim_{x \to 0} \\frac{\\sin(ax)}{x} = a$ for constant $a$.

  • Here, $a = 2$, so the limit is $2$.

Limits Involving Trigonometric Functions

Trigonometric limits often use identities and standard results.

• Example: $\\lim_{x \to 0} \\frac{1 - \\cos(5x)}{x}$

  • Use Taylor expansion: $\\cos(5x) \approx 1 - \\frac{(5x)^2}{2}$ for small $x$.

  • So, $1 - \\cos(5x) \approx \\frac{25x^2}{2}$, so the limit is $0$.

Continuity and Piecewise Functions

Continuity at a point requires that the function is defined, the limit exists, and the value matches the limit.

• Definition: A function $f(x)$ is continuous at $x = a$ if $\\lim_{x \to a} f(x) = f(a)$.

• Piecewise Functions: Functions defined by different expressions over different intervals. Check continuity at the boundaries.

Example: Piecewise Function

• Function: $g(x) = \\begin{cases} \\sqrt{x}, & x \geq 0 \\ \\ 1 - \\cos x, & x < 0 \\end{cases}$

• Continuity at $x = 0$:

  • Check $\\lim_{x \to 0^-} g(x)$ and $\\lim_{x \to 0^+} g(x)$.

  • If both limits equal $g(0)$, the function is continuous at $x = 0$.

Derivatives: Definition and Calculation

The derivative measures the instantaneous rate of change of a function. The formal definition uses limits.

• Definition: The derivative of $f(x)$ at $x = a$ is $f'(a) = \\lim_{h \to 0} \\frac{f(a + h) - f(a)}{h}$.

• Geometric Interpretation: The derivative at a point gives the slope of the tangent line to the curve at that point.

Example: Derivative Calculation

• Function: $f(x) = \\frac{2x}{x^2 + 1}$

• Derivative:

  • Apply the definition or use the quotient rule: $f'(x) = \\frac{(x^2 + 1) \\cdot 2 - 2x \\cdot 2x}{(x^2 + 1)^2}$

Table: Common Limits and Derivatives

Function	Limit as $x \to 0$	Derivative
$\\sin x$	$\\lim_{x \to 0} \\frac{\\sin x}{x} = 1$	$\\cos x$
$\\cos x$	$\\lim_{x \to 0} \\frac{1 - \\cos x}{x} = 0$	$-\\sin x$
$x^n$	$\\lim_{x \to a} x^n = a^n$	$n x^{n-1}$

Applications

• Tangent Line: The equation of the tangent line to $f(x)$ at $x = a$ is $y = f(a) + f'(a)(x - a)$.

• Continuity and Differentiability: A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability.

Summary

• Limits are essential for understanding the behavior of functions near specific points.

• Derivatives describe rates of change and are defined using limits.

• Piecewise functions require careful analysis for continuity and differentiability.

• Standard limit and derivative formulas are useful for quick calculations.

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard calculus curriculum.