Exploring Limits of Functions Using Tabular, Graphical, and Algebraic Methods

Tabular Method for Finding Limits

• Evaluates function values $f(x)$ as input values of $x$ closely approach a target number from both left and right directions.
• Example Functions for Tabular Analysis:
  • $f(x) = 3x^2 + x - 4$ evaluated near $x = 1$ and $x = -2$
  • $f(x) = \frac{x-1}{x+1}$ evaluated near $x = 0$ and $x = 1$

Graphical Method for Finding Limits

• Determines whether a limit exists and finds its value by observing the visual behavior of the function graph near a specific $x$-value.
• Evaluates whether the left-hand limit $\lim_{x \to a^-} f(x)$ and right-hand limit $\lim_{x \to a^+} f(x)$ approach the same target value.

Algebraic Method for Evaluating Limits

• Evaluates limits analytically using substitution or algebraic properties.
• Given Limit Problems:
  • $\lim_{x \to 1} (x^2 + 2x - 1)$
  • $\lim_{x \to -2} \frac{4}{6+x}$
  • $\lim_{x \to 0} (3x^2 - x + 5)$
  • $\lim_{x \to 1} \frac{x}{x+1}$
  • $\lim_{x \to 4} \frac{3}{x-4}$
  • $\lim_{x \to 2} f(x)$ where $f(x) = \begin{cases} 3x - 4 & x \le 2 \\ |x| & x > 2 \end{cases}$
  • $\lim_{x \to 2} (|x| - 3)$
  • $\lim_{x \to 3} (x - 3)(x + 2)$
  • $\lim_{x \to -1} \sqrt{x^2 + 4}$
  • $\lim_{x \to 3} (2x^2 - 5)$