1.6.1: Resources and Key Concepts
- Page ID
- 192190
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Resources
Videos
- Precise definition of a limit
- Epsilon-delta definition
- \(\epsilon-\delta\) proofs
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Proofs on Linear Functions
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Finding \( \delta \) Given \( \epsilon \)
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Proofs on Nonlinear Functions
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Proof: Functions w/Same Limits
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Example: sin(x)/x
- Limits - The Precise Definition of a Finite Limit at a Finite Point - Example: Disproving a Limit
- Proving limit laws
- One-sided limits (precise definition)
Key Concepts
Definitions
- \(\delta\)-Neighborhood Centered at \(a\): The interval \((a - \delta, a) \cup (a, a + \delta)\) on the \(x\)-axis. (Also referred to as a punctured open neighborhood).
- \(\epsilon\)-Neighborhood Centered at \(L\): The interval \((L - \epsilon, L + \epsilon)\) on the \(y\)-axis.
- Finite Limit at a Finite Number (Precise Definition): Let \(f(x)\) be defined for all \(x \neq a\) over an open interval containing \(a\), and let \(L\) be a real number. We say \( \displaystyle \lim_{x \to a} f(x) = L\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\), such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\).
- Symbolic Logic Statement: \(\forall \epsilon > 0, \exists \delta > 0 \ni 0 < |x-a| < \delta \Rightarrow |f(x)-L| < \epsilon\).
- Antecedent (in an "If..., then..." statement): The "If" part of the statement.
- Consequent (in an "If..., then..." statement): The "then" part of the statement.
- One-Sided Finite Limits at Finite Numbers (Precise Definition):
- Limit from the Right: Let \(f(x)\) be defined over an open interval of the form \((a, b)\) where \(a < b\). Then \( \displaystyle \lim_{x \to a^+} f(x) = L\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\), such that if \(0 < x - a < \delta\), then \(|f(x) - L| < \epsilon\).
- Limit from the Left: Let \(f(x)\) be defined over an open interval of the form \((b, c)\) where \(b < c\). Then \( \displaystyle \lim_{x \to c^-} f(x) = L\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\), such that if \(-\delta < x - c < 0\), then \(|f(x) - L| < \epsilon\).
Theorems
- Triangle Inequality: If \(a\) and \(b\) are any real numbers, then \(|a+b| \le |a|+|b|\).
Common Mistakes
- Symbolic Logic for "Such That": The symbol \(\ni\) is used frequently in symbolic logic for "such that"; however, many mathematicians do not use it for that purpose. Students should check with their instructor to see what logic symbol they prefer (many avoid the confusion entirely and use "s.t.").