1.2.1: Resources and Key Concepts

Resources

Videos

• Graphical limits
  • Limits - The Basic Concepts of Limits - Finite Limits at Finite Points
• One-sided limits from graph
  • Limits - The Basic Concepts of Limits - Single-Sided Limits
• Conceptual evaluation of limits
  • Limits - The Basic Concepts of Limits - Investigating Limits Conceptually

Key Concepts

Definitions

• Vertical Asymptote: The function \(f(x)\) is said to have a vertical asymptote at \(x = a\) if \( \displaystyle \lim_{x \to a^-} f(x) = \pm\infty\), or \( \displaystyle \lim_{x \to a^+} f(x) = \pm\infty\), or \( \displaystyle \lim_{x \to a} f(x) = \pm\infty\).

Theorems

• Theorem (Conceptual Investigation of an Infinite Limit): Let \(f(x) = \frac{N(x)}{D(x)}\), where \( \displaystyle \lim_{x \to a} N(x)\) is a finite, nonzero constant, and \(D(x)\) is approaching \(0\) as \(x\) approaches \(a\). Then \( \displaystyle \lim_{x \to a} \frac{N(x)}{D(x)}\) is either \(\infty\) or \(-\infty\). The sign is determined by comparing the signs of the numerator and denominator of the ratio as \(x\) is approaching \(a\).

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 the function values as \(x\) gets closer to \(a\), not what \(f(a)\) is. This is especially clear graphically where there might be an open circle at \(x=a\).
• Caution: Issues with Graphical Limit Investigations:
  • If not provided with a graph, creating an accurate one can be difficult or impossible.
  • Graphs can be visually "tricky," potentially hiding details like an open circle (where the function is undefined) or leading to incorrect assumptions about function behavior if not interpreted carefully. (e.g., the open circle at \((0,1)\) for \(f(\theta) = \sin\theta/\theta\)).
• Caution: Calculus Does Not Work in Degrees: When working with trigonometric functions in Calculus, angles must be expressed in radians, not degrees. This affects the appearance of their graphs and thus limit investigations.
• Caution: Limits are Unconcerned with \(f(a)\): The value of \( \displaystyle \lim_{x \to a} f(x)\) depends on the function's behavior near \(a\), not at \(a\). The function might be undefined at \(a\), or \(f(a)\) might be different from the limit.
• Caution: Graphical Observation is Not Proof: Observing that a graph appears to approach a certain value (e.g., \( \displaystyle \lim_{\theta \to 0} \frac{\sin\theta}{\theta}\) appears to be 1 from its graph) is an estimation, not a formal proof.
• Caution: Never Write That a Limit is Equal to an Indeterminate Form: When conceptually evaluating limits that tend to forms like \(\frac{k}{0}\) (where \(k \neq 0\)), do not write that the limit equals \(\frac{k}{0^+}\) or \(\frac{k}{0^-}\). These expressions (\(\frac{k}{0^+}\), etc.) describe tendencies, not actual numbers. Use arrow notation to show the behavior of the numerator and denominator separately, then conclude the limit (e.g., \( \displaystyle \lim_{x \to 7^+} \frac{2}{x-7}\). Numerator \(\to 2\), Denominator \((x-7) \to 0^+\). Thus, \( \displaystyle \lim_{x \to 7^+} \frac{2}{x-7} = \infty\)).