Calculus I Exam Study Guide: Limits, Continuity, and Tangents

Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives, continuity, and analyzing function behavior near points of interest.

• 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: $\lim_{x \to a} f(x)$

• Key Properties:

  • If the left-hand and right-hand limits are equal, the limit exists.

  • Limits can be finite or infinite, and may not exist if the function behaves erratically near the point.

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

Evaluating Limits

Limits can be evaluated using algebraic manipulation, substitution, and special limit laws. Some problems require recognizing indeterminate forms and applying techniques such as factoring, rationalizing, or using L'Hôpital's Rule.

• Direct Substitution: If f(x) is continuous at x = a, then $\lim_{x \to a} f(x) = f(a)$.

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

• Rationalizing: Multiply by a conjugate to simplify square roots.

• Special Trigonometric Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• Example: $\lim_{x \to 2} \frac{\sqrt{x+2} - 2}{x-2}$ (rationalize numerator)

Continuity

A function is continuous at a point if its limit at that point equals its value. Continuity is crucial for many calculus theorems and for defining derivatives.

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

• Interval Notation: Used to describe where a function is continuous.

• Types of Discontinuity:

  • Removable: The limit exists, but f(a) is undefined or not equal to the limit.

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

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is discontinuous at x = 1, but the discontinuity is removable.

Average and Instantaneous Rates of Change

Average Speed

The average rate of change of a function over an interval is the change in output divided by the change in input. For position functions, this represents average speed.

• Formula: $\text{Average speed} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}$

• Example: For $h(t) = -16t^2 + 2880$, average speed from $t = 0$ to $t = 60$ is $\frac{h(60) - h(0)}{60 - 0}$

Instantaneous Speed and Tangent Lines

The instantaneous rate of change at a point is the derivative, representing the slope of the tangent line to the curve at that point.

• Definition: $h'(t) = \lim_{\Delta t \to 0} \frac{h(t + \Delta t) - h(t)}{\Delta t}$

• Tangent Line Equation: $y = h'(a)(x - a) + h(a)$

• Example: Find $h'(1)$ for $h(t) = -16t^2 + 2880$ and write the tangent line at $t = 1$.

Solving Limits: Practice Problems

Algebraic and Trigonometric Limits

• Example 1: $\lim_{x \to \frac{\pi}{2}} (\sin(x/2) - 3\cos(x))$

• Example 2: $\lim_{x \to 0} \frac{\sin(5x)}{x}$

• Example 3: $\lim_{x \to 4} \frac{\sqrt{x+2} - 2}{x-4}$

Limits Involving Rational Functions

• Example 1: $\lim_{x \to 3} \frac{7x - 21}{x - 3}$

• Example 2: $\lim_{x \to 2} \frac{x^2 + 1}{x^2 - 5x + 6}$

• Example 3: $\lim_{x \to 1} \frac{7x - 3}{\sqrt{x^2 + 1}}$

Continuity and Discontinuity

Describing Continuity

To describe where a function is continuous, use interval notation and analyze points where the function may be undefined or discontinuous.

• Interval Notation: Expresses all x-values where the function is continuous, e.g., (a, b), [a, b], or unions of intervals.

• Example: $f(x) = \frac{1}{x-2}$ is continuous everywhere except at $x = 2$.

Types of Discontinuity

• Removable Discontinuity: The limit exists, but the function is not defined or not equal to the limit at that point.

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

• Using Limits to Classify: Analyze $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ to determine the type.

Existence of Solutions and the Intermediate Value Theorem

Intermediate Value Theorem (IVT)

The IVT states that if a function is continuous on [a, b] and takes values f(a) and f(b), then it takes every value between f(a) and f(b) at some point in (a, b).

• Application: Used to show that equations like $f(x) = 0$ have solutions in an interval.

• Example: For $f(x) = x^3 - 5x^2 - 10x^2 + 15x + 30$, check if $f(x) = 0$ has a solution in [2, 3] by evaluating $f(2)$ and $f(3)$.

Special Limit Inequalities

Absolute Value and Trigonometric Inequalities

Limits involving inequalities and absolute values often require careful analysis, especially near points of discontinuity or where the function is not defined.

• Example: $-1 \leq \sin\left(\frac{2}{x}\right) \leq 1$ for $x$ in $[-1, 1]$, except $x = 0$.

• Application: Use this fact to evaluate $\lim_{x \to 0} (\cos(2/x))$.

Summary Table: Types of Discontinuity

Type	Description	Example
Removable	Limit exists, function value missing or not equal to limit	$f(x) = \frac{x^2 - 1}{x - 1}$ 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 near the point	$f(x) = \frac{1}{x}$ at $x = 0$

Practice Problem Summary

• Evaluate limits using algebraic and trigonometric techniques.

• Determine intervals of continuity and classify discontinuities.

• Apply the Intermediate Value Theorem to prove existence of solutions.

• Find average and instantaneous rates of change for position functions.

• Write equations of tangent lines using derivatives.

Additional info: The study notes expand on exam questions by providing definitions, examples, and context for each calculus topic covered, including limits, continuity, rates of change, and tangent lines.