Comprehensive Guide to Unit 1: Limits and Continuity

Limit

The value that a function f(x) approaches as the input value x gets closer to some number c.

Formal Notation of a Limit

Written as ( \lim_{x \to c} f(x) = L ), meaning the limit of f(x) as x approaches c equals L.

Left-Hand Limit

Approaching c from values smaller than c, denoted as ( \lim_{x \to c^-} f(x) ).

Right-Hand Limit

Approaching c from values larger than c, denoted as ( \lim_{x \to c^+} f(x) ).

General Limit Existence Theorem

( \lim{x \to c} f(x) = L ) if and only if ( \lim{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L ).

Does Not Exist (DNE)

The condition that occurs when the left and right limits do not match.

Graphical Estimation

Using a graph to trace the function and analyze its limits as x approaches a certain value.

Holes in Graph

Points where the limit exists but f(c) is not defined, leading to a discontinuity.

Jumps in Graph

Points where the limit does not exist due to a break in the graph.

Vertical Asymptotes

Occurs when a function shoots up or down infinitely as it approaches a specific x-value.

Direct Substitution

The first step in determining limits by plugging the value c into the function.

Indeterminate Form

A condition represented as ( \frac{0}{0} ) that requires further algebraic manipulation to find a limit.

Factoring and Canceling

A technique used when an indeterminate form arises to simplify the function before substituting again.

Rationalization

Multiplying by the conjugate to simplify limits involving square roots.

Complex Fractions

Fractions that contain other fractions and need a common denominator to simplify.

Special Trigonometric Limits

Memorizing limits such as ( \lim{x \to 0} \frac{\sin x}{x} = 1 ) and ( \lim{x \to 0} \frac{1 - \cos x}{x} = 0 ) as x approaches 0.

Squeeze Theorem

A method to evaluate limits by 'squeezing' a function between two other functions.

Removable Discontinuity

Occurs when the limit exists but f(c) is either undefined or differs from the limit.

Jump Discontinuity

When the left-hand limit does not equal the right-hand limit at a certain point.

Infinite Discontinuity

One or both one-sided limits approach ( \pm\infty ), indicating a vertical asymptote.

The 3-Step Test for Continuity

1. f(c) is defined. 2. ( \lim{x \to c} f(x) ) exists. 3. ( \lim{x \to c} f(x) = f(c) ).

Extended Function

The new function created by simplifying to remove a removable discontinuity.

Vertical Asymptote (Infinite Limits)

A line x = c where ( \lim_{x \to c} f(x) = \pm\infty ).

Horizontal Asymptote (Limits at Infinity)

A line y = L where ( \lim{x \to \infty} f(x) = L ) or ( \lim{x \to -\infty} f(x) = L ).

Bottom Heavy Rational Functions

When the degree of P(x) is less than that of Q(x), leading to a limit of 0 and HA at y=0.

Equal Degree Rational Functions

When the degrees are equal, the limit is the ratio of leading coefficients for horizontal asymptotic behavior.

Top Heavy Rational Functions

When the degree of P(x) is greater than that of Q(x), leading to a limit of ( \pm \infty ), indicating no horizontal asymptote.

End Behavior of Functions

The description of a function as x approaches infinity or negative infinity.

Intermediate Value Theorem (IVT)

States that if f(x) is continuous on [a, b], then for any k between f(a) and f(b), there is at least one c in (a, b) such that f(c) = k.

Existence Theorem

Another term for the Intermediate Value Theorem, indicating the existence of roots or outputs.

Real World Analogy for IVT

If you grew continuously from 3 feet to 5 feet, there must have been a moment when you were exactly 4 feet tall.

Common Mistake: The '0/0' Answer

Never write '0/0' as a final answer; it indicates further work is needed.

Common Mistake: Limit vs. Function Value

A limit describes what happens as x approaches c, not necessarily the value of the function at c.

Common Mistake: One-Sided Limits notations

Notation matters; ( \lim{x \to 2^-} ) is not the same as ( \lim{x \to -2} ).

Common Mistake: Conditions for IVT

The function must be continuous on the interval to apply the Intermediate Value Theorem.

Common Mistake: Substituting Infinity

Infinity is not a number; do not perform arithmetic directly with it.