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Essential Concepts

The Limit Laws

• The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time.
• For polynomials and rational functions, [latex]\underset{x\to a}{\lim}f(x)=f(a)[/latex].
• You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction.
• The Squeeze Theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known.

Continuity

• For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.
• Discontinuities may be classified as removable, jump, or infinite.
• A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
• The composite function theorem states: If [latex]f(x)[/latex] is continuous at [latex]L[/latex] and [latex]\underset{x\to a}{\lim}g(x)=L[/latex], then [latex]\underset{x\to a}{\lim}f(g(x))=f(\underset{x\to a}{\lim}g(x))=f(L)[/latex].
• The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.

The Precise Definition of a Limit

• The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit.
• The epsilon-delta definition may be used to prove statements about limits.
• The epsilon-delta definition of a limit may be used to find deltas algebraically.

Key Equations

• Basic Limit Results
  [latex]\underset{x\to a}{\lim}x=a[/latex]
  [latex]\underset{x\to a}{\lim}c=c[/latex]
• Important Limits
  [latex]\underset{\theta \to 0}{\lim} \sin \theta =0[/latex]
  [latex]\underset{\theta \to 0}{\lim} \cos \theta =1[/latex]
  [latex]\underset{\theta \to 0}{\lim}\frac{\sin \theta}{\theta}=1[/latex]
  [latex]\underset{\theta \to 0}{\lim}\frac{1- \cos \theta}{\theta}=0[/latex]

Glossary

continuity at a point

A function [latex]f(x)[/latex] is continuous at a point [latex]a[/latex] if and only if the following three conditions are satisfied: (1) [latex]f(a)[/latex] is defined, (2) [latex]\underset{x\to a}{\lim}f(x)[/latex] exists, and (3) [latex]\underset{x\to a}{\lim}f(x)=f(a)[/latex]

continuity from the left

A function is continuous from the left at [latex]b[/latex] if [latex]\underset{x\to b^-}{\lim}f(x)=f(b)[/latex]

continuity from the right

A function is continuous from the right at [latex]a[/latex] if [latex]\underset{x\to a^+}{\lim}f(x)=f(a)[/latex]

continuity over an interval

a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function [latex]f(x)[/latex] is continuous over a closed interval of the form [latex][a,b][/latex] if it is continuous at every point in [latex](a,b)[/latex], and it is continuous from the right at [latex]a[/latex] and from the left at [latex]b[/latex]

discontinuity at a point

A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point

epsilon-delta definition of the limit

[latex]\underset{x\to a}{\lim}f(x)=L[/latex] if for every [latex]\varepsilon >0[/latex], there exists a [latex]\delta >0[/latex] such that if [latex]0<|x-a|<\delta[/latex], then [latex]|f(x)-L|<\varepsilon[/latex]

infinite discontinuity

An infinite discontinuity occurs at a point [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)=\pm \infty[/latex] or [latex]\underset{x\to a^+}{\lim}f(x)=\pm \infty[/latex]

infinite limit

A function has an infinite limit at a point [latex]a[/latex] if it either increases or decreases without bound as it approaches [latex]a[/latex]

Intermediate Value Theorem

Let [latex]f[/latex] be continuous over a closed bounded interval [latex][a,b][/latex]; if [latex]z[/latex] is any real number between [latex]f(a)[/latex] and [latex]f(b)[/latex], then there is a number [latex]c[/latex] in [latex][a,b][/latex] satisfying [latex]f(c)=z[/latex]

intuitive definition of the limit

If all values of the function [latex]f(x)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x(\ne a)[/latex] approach [latex]a[/latex], [latex]f(x)[/latex] approaches [latex]L[/latex]

jump discontinuity

A jump discontinuity occurs at a point [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)[/latex] both exist, but [latex]\underset{x\to a^-}{\lim}f(x) \ne \underset{x\to a^+}{\lim}f(x)[/latex]

one-sided limit

A one-sided limit of a function is a limit taken from either the left or the right

removable discontinuity

A removable discontinuity occurs at a point [latex]a[/latex] if [latex]f(x)[/latex] is discontinuous at [latex]a[/latex], but [latex]\underset{x\to a}{\lim}f(x)[/latex] exists

triangle inequality

If [latex]a[/latex] and [latex]b[/latex] are any real numbers, then [latex]|a+b|\le |a|+|b|[/latex]

vertical asymptote

A function has a vertical asymptote at [latex]x=a[/latex] if the limit as [latex]x[/latex] approaches [latex]a[/latex] from the right or left is infinite

Study Tips

Evaluating Limits

• Practice applying limit laws to simplify complex expressions.
• For rational functions, check if direct substitution works before using advanced techniques.
• When encountering [latex]\frac{0}{0}[/latex], try factoring or simplifying first.
• For limits with square roots, consider using the conjugate method.
• In complex fractions, find a common denominator before simplifying.
• Always check for potential cancelations after each step of simplification.
• Sketch graphs to visualize the behavior of functions near the limit point.

The Squeeze Theorem

• Practice identifying appropriate bounding functions for the Squeeze Theorem.
• Memorize the key trigonometric limits as they are foundational for calculus.
• Use the unit circle to understand the behavior of trigonometric functions near zero.
• When evaluating complex trigonometric limits, try to manipulate the expression to use known limits.
• Remember that these limits apply to angles in radians, not degrees.

Continuity

• Graph functions to visualize continuity and discontinuities.
• For piecewise functions, check continuity at the transition points.
• Remember that rational functions are only discontinuous where the denominator is zero.
• Practice finding domains of rational functions to identify potential discontinuities.

Types of Discontinuities

• For rational functions, factor the numerator and denominator to spot potential removable discontinuities.
• Remember that removable discontinuities can be “fixed” by redefining the function at that point.
• Use one-sided limits to distinguish between jump and removable discontinuities.
• Look for denominators that equal zero to identify potential infinite discontinuities.
• Practice classifying discontinuities in piecewise functions, especially at transition points.

Continuity Over an Interval

• Practice identifying the domain of functions to determine potential intervals of continuity.
• For rational functions, focus on points where the denominator could be zero.
• With radical functions, consider the domain restrictions carefully.
• For piecewise functions, check continuity at transition points within the interval.
• Remember that polynomials are continuous over their entire domain (all real numbers).

Composite Function Theorem and The Intermediate Value Theorem

• Practice applying the Composite Function Theorem to solve complex limit problems.
• Visualize the Intermediate Value Theorem using graphs to understand its implications.
• When using IVT to prove existence of zeros, focus on finding intervals where the function changes sign.
• Remember that continuity is a crucial condition for both theorems.

Epsilon-Delta Definition of the Limit

• Practice visualizing the [latex]\varepsilon[/latex]–[latex]\delta[/latex] relationship graphically.
• Pay attention to how [latex]\delta[/latex] is chosen based on [latex]\varepsilon[/latex] and the function.
• Remember that [latex]\delta[/latex] can depend on [latex]\varepsilon[/latex], but not on [latex]x[/latex].
• Use algebraic manipulation to relate [latex]|x - a|[/latex] to [latex]|f(x) - L|[/latex].

Advanced Applications of the Epsilon-Delta Definition: Proofs, Non-Existence, and Algebraic Calculations

• Practice using the triangle inequality in epsilon-delta proofs.
• For non-existence proofs, think about how the function behaves near the point of interest.
• In algebraic delta calculations, solve inequalities step-by-step.

One-Sided and Infinite Limits

• Compare one-sided limit definitions with the standard two-sided definition.
• Practice proving one-sided limits using the modified epsilon-delta definition.
• When working with infinite limits, think about how to show a function grows arbitrarily large.
• Remember that one-sided limits are crucial for understanding function behavior at discontinuities.