Question

Please answer and explain the link below

Answer 1

Answer:

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]:                                                             $\displaystyle \lim_{x \to c} x = c$

Step-by-step explanation:

Step 1: Define

Identify

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

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]:                                                       $\displaystyle \lim_{x \to 3^-} \sqrt{x + 1}$
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           $\displaystyle \lim_{x \to 3^-} \sqrt{x + 1} = \sqrt{3 + 1}$
3. [√Radical] Add:                                                                                             $\displaystyle \lim_{x \to 3^-} \sqrt{x + 1} = \sqrt{4}$
4. [√Radical] Evaluate:                                                                                       $\displaystyle \lim_{x \to 3^-} \sqrt{x + 1} = 2$

∴ 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