Question

Please answer and explain the link below

Answer 1

See explanation.

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Functions
• Function Notation

Calculus

Limits

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

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

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = \left \{ {{√(x + 1), \ x < 3} \atop {5 - x, \ x \geq 3}} \right.`

Step 2: Find Right Limit

1. Substitute in variables [Right-Side Limit]:
   `\displaystyle \lim_(x \to 3^+) 5 - x`
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle \lim_(x \to 3^+) 5 - x = 5 - 3`
3. Subtract:
   `\displaystyle \lim_(x \to 3^+) 5 - x = 2`

∴ the right-side limit equals 2.

Step 3: Find Left Limit

1. Substitute in variables [Left-Side Limit]:
   `\displaystyle \lim_(x \to 3^-) √(x + 1)`
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle \lim_(x \to 3^-) √(x + 1) = √(3 + 1)`
3. [√Radical] Add:
   `\displaystyle \lim_(x \to 3^-) √(x + 1) = √(4)`
4. [√Radical] Evaluate:
   `\displaystyle \lim_(x \to 3^-) √(x + 1) = 2`

∴ the left-side limit equals 2.

Step 4: Find Limit

The right and left-side limits are equal.

∴
`\displaystyle \lim_(x \to 3) f(x) = 2`

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

Unit: Limits

Book: College Calculus 10e