1.1.1: Resources and Key Concepts

Resources

Videos

• Limit of a function
  • Limits - The Basic Concepts of Limits - Finite Limits at Finite Points
• One-sided limits
  • Limits - The Basic Concepts of Limits - Single-Sided Limits

Key Concepts

Definitions

• Finite Limit at a Finite Number (Intuitive Definition): Let \(f(x)\) be a function defined at all values in an open interval containing a finite number \(a\), with the possible exception of \(a\) itself, and let \(L\) be a finite real number. If the values of the function \(f(x)\) approach the real number \(L\) as the values of \(x\) (where \(x \neq a\)) approach the number \(a\), then we say that the finite limit of \(f(x)\) as \(x\) approaches the finite number \(a\) is \(L\). Symbolically, \( \displaystyle \lim_{x \to a} f(x) = L\).
• DNE (Notation): An abbreviation for "does not exist," used to indicate that a limit does not approach a single value (whether finite or infinity).
• Limit from the left: Let \(f(x)\) be a function defined at all values in an open interval of the form \((z, a)\), and let \(L\) be a real number. If the values of \(f(x)\) approach \(L\) as the values of \(x\) (where \(x < a\)) approach \(a\), then \(L\) is the limit of \(f(x)\) as \(x\) approaches \(a\) from the left, written \( \displaystyle \lim_{x \to a^-} f(x) = L\).
• Limit from the right: Let \(f(x)\) be a function defined at all values in an open interval of the form \((a, c)\), and let \(L\) be a real number. If the values of \(f(x)\) approach \(L\) as the values of \(x\) (where \(x > a\)) approach \(a\), then \(L\) is the limit of \(f(x)\) as \(x\) approaches \(a\) from the right, written \( \displaystyle \lim_{x \to a^+} f(x) = L\).
• Infinite limits from the left: If the values of \(f(x)\) increase without bound as \(x\) approaches \(a\) from the left (\(x < a\)), the limit is positive infinity (\( \displaystyle \lim_{x \to a^-} f(x) = +\infty\)). If \(f(x)\) decrease without bound, the limit is negative infinity (\( \displaystyle \lim_{x \to a^-} f(x) = -\infty\)).
• Infinite limits from the right: If the values of \(f(x)\) increase without bound as \(x\) approaches \(a\) from the right (\(x > a\)), the limit is positive infinity (\( \displaystyle \lim_{x \to a^+} f(x) = +\infty\)). If \(f(x)\) decrease without bound, the limit is negative infinity (\( \displaystyle \lim_{x \to a^+} f(x) = -\infty\)).
• Two-sided infinite limit: If the values of \(f(x)\) increase without bound as \(x\) (where \(x \neq a\)) approaches \(a\), the limit is positive infinity (\( \displaystyle \lim_{x \to a} f(x) = +\infty\)). If \(f(x)\) decrease without bound, the limit is negative infinity (\( \displaystyle \lim_{x \to a} f(x) = -\infty\)).

Theorems

• Theorem: Two Important Limits

  Let \(a\) be a real number and \(c\) be a constant.

  1. \( \displaystyle \lim_{x \to a} x = a\) (Limit of the Identity Function)
  2. \( \displaystyle \lim_{x \to a} c = c\) (Limit of a Constant)
• Theorem: Relating One-Sided and Two-Sided Limits

  Let \(f(x)\) be a function defined at all values in an open interval containing \(a\), with the possible exception of \(a\) itself, and let \(L\) be a real number, \(\infty\), or \(-\infty\). Then, \[ \lim_{x \to a} f(x) = L \nonumber \] if and only if \[ \lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L. \nonumber \]

Common Mistakes

• Caution: \(x \neq a\): The intuitive definition of a finite limit at a finite number states that \(x\) need not equal \(a\). The limit is concerned with function values as \(x\) gets closer to \(a\), not what \(f(a)\) is. It could be that \(f(a)\) does not exist or \(L \neq f(a)\).
• Caution: Be prepared to try "wiggle" sizes other than powers of ten: While choosing \(x\)-values like \(a \pm 0.1, a \pm 0.01\), etc., often works, one may need to modify the choice of "wiggle" sizes (the difference from \(a\)) on some occasions to better observe the limiting behavior.
• Caution: Calculus Does Not Work in Degrees: When working with trigonometric functions in Calculus, angles must be expressed in radians, not degrees.
• Caution: Observations Are Not Proofs: Observing a trend in a table of function values that suggests a limit (e.g., that \( \displaystyle \lim_{x \to a} c\) looks to be \(c\)) is not a mathematical proof. Formal proofs are required for such statements.
• Caution: Using Tables for Limits is a Terrible Idea: Attempting to estimate a limit using a table of values outside of introductory exercises in this section is highly discouraged. This method can be misleading due to finite precision arithmetic or by not choosing appropriate \(x\)-values.