Calculus Study Guide: Limits, Continuity, and Differentiation

Limits and Continuity

Understanding Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. They are essential for defining derivatives and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a.

• Notation: $\lim_{x \to a} f(x)$

• Existence: A limit exists if the left-hand and right-hand limits are equal.

• Graphical Interpretation: Limits can be estimated using the graph of the function by observing the y-value as x approaches the target value.

Example: If $f(x)$ approaches 2 as $x$ approaches 1, then $\lim_{x \to 1} f(x) = 2$.

Limits from Graphs

Graphs are useful for visually determining limits, especially when the function is not defined algebraically at the point of interest.

• Look for the y-value the function approaches as x nears the target value.

• If the graph jumps or has a hole, the limit may not exist.

Example: If the graph approaches y = 3 from both sides as x approaches 2, then $\lim_{x \to 2} f(x) = 3$.

Continuity

A function is continuous at a point if the limit exists and equals the function's value at that point.

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

• Discontinuities occur at jumps, holes, or vertical asymptotes.

Example: The function $f(x)$ is continuous over the interval (-6, 6) if there are no breaks in the graph within that interval.

Techniques for Evaluating Limits

Algebraic Methods

Limits can often be evaluated by direct substitution, factoring, or rationalizing.

• Direct Substitution: Substitute the value of x into the function.

• Factoring: Factor the numerator and denominator to cancel common terms.

• Rationalizing: Multiply by a conjugate to simplify expressions with roots.

Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ Factor numerator: $x^2 - 4 = (x - 2)(x + 2)$ Cancel $(x - 2)$: $\lim_{x \to 2} x + 2 = 4$

Theorem on Limits of Rational Functions

For rational functions, if direct substitution yields a nonzero denominator, the limit can be found by substituting the value directly.

• Example: $\lim_{x \to 2} \frac{6x^2 - 8}{6x - 2}$ Substitute x = 2: $\frac{6(2)^2 - 8}{6(2) - 2} = \frac{24 - 8}{12 - 2} = \frac{16}{10} = \frac{8}{5}$

Differentiation

Definition of the Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the difference quotient.

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

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

Example: For $f(x) = x^2$, $f'(x) = 2x$.

Basic Differentiation Rules

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Constant Multiple Rule: $\frac{d}{dx} [c f(x)] = c f'(x)$

• Sum Rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

Example: $\frac{d}{dx} (3x^4) = 12x^3$

Product and Quotient Rules

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Example: $\frac{d}{dx} \left( \frac{8x^2 - 3}{7x^5} \right)$

Chain Rule

The chain rule is used to differentiate composite functions.

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Example: $\frac{d}{dx} (x^2 + 4)^5 = 5(x^2 + 4)^4 \cdot 2x = 10x(x^2 + 4)^4$

Applications of Derivatives

Velocity and Acceleration

Derivatives are used to describe rates of change in physical contexts, such as velocity and acceleration.

• Velocity: The derivative of position with respect to time.

• Acceleration: The derivative of velocity with respect to time.

Example: If $s(t) = t^2$, then $v(t) = \frac{ds}{dt} = 2t$ and $a(t) = \frac{dv}{dt} = 2$.

Marginal Analysis in Economics

Derivatives are used to find marginal cost, marginal revenue, and rates of change in business applications.

• Marginal Revenue: The derivative of the revenue function with respect to quantity.

• Example: If $R(x) = 100x - x^2$, then $R'(x) = 100 - 2x$.

Logarithmic and Exponential Functions

Properties and Graphs

Logarithmic and exponential functions are important in calculus for modeling growth and decay.

• Exponential Function: $f(x) = a^x$

• Logarithmic Function: $f(x) = \ln(x)$

• Domain: For $\ln(x)$, domain is $x > 0$.

Example: The graph of $y = \ln(x + 7)$ is defined for $x > -7$.

Summary Table: Key Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx} (x^2 \cdot \sin x)$
Quotient Rule	$\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx} \left( \frac{x^2}{x+1} \right)$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx} (\sin(x^2))$

Additional info:

• Some questions involve interpreting graphs to estimate limits and derivatives.

• Applications include population growth, marginal analysis, and motion problems.

• Logarithmic and exponential equations are solved using properties of logarithms and exponentials.