Math Problem Statement

Estimate the limits of functions g(x), sqrt(x-4), a rational function, and a piecewise function as x approaches specific points using tables of values.

Solution

The assignment contains four limit problems requiring estimates using tables of values:

  1. Problem 1: Estimate limx1g(x)\lim_{x \to 1} g(x) where g(x)=5x+8g(x) = 5x + 8, using tables of values.
  2. Problem 2: Estimate limx13x4\lim_{x \to 13} \sqrt{x - 4}, using tables of values.
  3. Problem 3: Estimate limx1x2+3x+2x+1\lim_{x \to -1} \frac{x^2 + 3x + 2}{x + 1}, using tables of values.
  4. Problem 4: For the piecewise function \begin{cases} 2x + 3, & \text{if } x < 1 \\ x^2 - 4, & \text{if } x \geq 1 \end{cases}$$ Estimate $$\lim_{x \to 1^-} t(x)$$ using a table of values.

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Math Problem Analysis

Mathematical Concepts

Limits
Piecewise Functions
Square Roots
Rational Functions
Continuity and Approaching Values

Formulas

Limit of a function as x approaches a point
g(x) = 5x + 8
sqrt(x - 4)
Rational function: (x^2 + 3x + 2) / (x + 1)
Piecewise function t(x) = { 2x+3 if x < 1, x^2-4 if x >= 1 }

Theorems

Definition of a Limit
Left-Hand and Right-Hand Limits
Simplification Techniques for Rational Functions

Suitable Grade Level

High School (Grades 10-12)