Calculus I: Limits, Continuity, and Introduction to Derivatives – Study Guide for Test 1

Limits and Continuity

Definitions and Theorems

This section covers foundational concepts in calculus, including the definition of a limit, types of discontinuities, and theorems related to limits and continuity.

• Limit of a Function: The value that a function approaches as the input approaches a certain point. Formally, $\lim\limits_{x \to a} f(x) = L$ means that as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$.

• One-Sided Limits: The limit of $f(x)$ as $x$ approaches $a$ from the left ($x \to a^-$) or right ($x \to a^+$).

• Continuity at a Point: A function $f(x)$ is continuous at $x = a$ if $\lim\limits_{x \to a} f(x) = f(a)$.

• Types of Discontinuities:

  • Removable Discontinuity: The limit exists, but $f(a)$ is either not defined or not equal to the limit.

  • Non-removable Discontinuity: The limit does not exist due to a jump or infinite discontinuity.

• Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

Evaluating Limits

Limits can be evaluated using various methods, including tables, graphs, algebraic manipulation, and the Squeeze Theorem.

• Table of Values: Substitute values close to the point of interest to estimate the limit.

• Graphical Approach: Observe the behavior of the function near the point on its graph.

• Algebraic Methods: Simplify the function using factoring, rationalization, or other algebraic techniques to evaluate the limit.

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$ (except possibly at $a$), and $\lim\limits_{x \to a} g(x) = \lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} f(x) = L$.

Limits Involving Transcendental Functions

Transcendental functions include exponential, logarithmic, and trigonometric functions. Limits involving these functions often require special techniques or knowledge of their properties.

• Example: $\lim\limits_{x \to 0} \sin(x)/x = 1$

• Example: $\lim\limits_{x \to \infty} e^{-x} = 0$

Piecewise Functions and Continuity

Piecewise functions are defined by different expressions over different intervals. To analyze their limits and continuity:

• Check the limit from the left and right at the points where the formula changes.

• Determine if the function is continuous at those points by comparing the left and right limits to the function value.

Discontinuities and Intervals of Continuity

To determine where a function is continuous or has discontinuities:

• Find points where the function is not defined or where the limit does not exist.

• Classify discontinuities as removable or non-removable.

• State the intervals where the function is continuous.

Intermediate Value Theorem (IVT)

The IVT is used to show that a function takes on every value between $f(a)$ and $f(b)$ on a continuous interval $[a, b]$.

• Check that the function is continuous on the interval.

• Verify that the value in question lies between $f(a)$ and $f(b)$.

• Conclude that there exists a $c$ in $(a, b)$ such that $f(c)$ equals the desired value.

Introduction to Derivatives

Limit Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point.

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

• Alternative Definition: $f'(a) = \lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a}$

• Equation of the Tangent Line: $y = f'(a)(x - a) + f(a)$

Difference Quotient

The difference quotient is used to compute the average rate of change and is foundational for the definition of the derivative.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• As $h \to 0$, the difference quotient approaches the derivative.

Velocity as a Derivative

For a position function $s(t)$, the velocity at time $t$ is the derivative $s'(t)$.

• Formula: $v(t) = \lim\limits_{h \to 0} \frac{s(t+h) - s(t)}{h}$

• Represents the instantaneous velocity at time $t$.

Graphical Interpretation of the Derivative

The graph of the derivative $f'(x)$ provides information about the slope of the original function $f(x)$ at each point.

• Where $f(x)$ is increasing, $f'(x) > 0$.

• Where $f(x)$ is decreasing, $f'(x) < 0$.

• Where $f(x)$ has a horizontal tangent, $f'(x) = 0$.

Sample Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, but function value is missing or different	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values on each side
Infinite	Function approaches infinity at a point	$f(x) = \frac{1}{x}$ at $x = 0$

Key Study Strategies

• Memorize definitions and theorems related to limits and continuity.

• Practice evaluating limits using tables, graphs, and algebraic methods.

• Be able to classify discontinuities and determine intervals of continuity.

• Understand and apply the Intermediate Value Theorem.

• Use the limit definition of the derivative to find tangent lines and instantaneous rates of change.

• Interpret and sketch the graph of a derivative given the graph of a function.

Additional info: These notes synthesize the test outline into a comprehensive study guide, expanding on brief points with academic context and examples.