Question

Given f(x) = 3sqrt(2x-1).
6(2x-1)^2-3

What is lim f(x)?

Answer 1

`\displaystyle 51`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Factoring
• Functions
• Function Notation

Algebra II

• Piecewise functions

Calculus

Limits

• Right-Side Limit:
  `\displaystyle \lim_(x \to c^+) f(x)`

Limit Rule [Variable Direct Substitution]:
`\displaystyle \lim_(x \to c) x = c`

Limit Property [Addition/Subtraction]:
`\displaystyle \lim_(x \to c) [f(x) \pm g(x)] = \lim_(x \to c) f(x) \pm \lim_(x \to c) g(x)`

Limit Property [Multiplied Constant]:
`\displaystyle \lim_(x \to c) bf(x) = b \lim_(x \to c) f(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = \left \{ {{3√(2x - 1), \ x \leq 2} \atop {6(2x - 1)^2 - 3, \ x > 2}} \right.`

Step 2: Solve

1. Substitute in function [Limit]:
   `\displaystyle \lim_(x \to 2^+) 6(2x - 1)^2 - 3`
2. Factor:
   `\displaystyle \lim_(x \to 2^+) 3[2(2x - 1)^2 - 1]`
3. Rewrite [Limit Property - Multiplied Constant]:
   `\displaystyle 3\lim_(x \to 2^+) 2(2x - 1)^2 - 1`
4. Evaluate [Limit Property - Variable Direct Substitution]:
   `\displaystyle 3[2(2 \cdot 2 - 1)^2 - 1]`
5. Simplify:
   `\displaystyle 51`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

Book: College Calculus 10e