Question

15 points!
Find the indicated limit, if it exists.​

Answer 1

The limit does not exist

General Formulas and Concepts:

Calculus

Limits

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

Limit Rule [Constant]:
`\displaystyle \lim_(x \to c) b = b`

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)`

Explanation:

*Note:

In order for the limit to exist, the right-hand and left-hand limits must equal each other

Step 1: Define

Identify

`\displaystyle f(x) = \left \{ {{x + 10,\ x < 8} \atop {10 - x,\ x \geq 8}} \right.`

Step 2: Find Right-Hand Limit

1. Substitute in function [Limit]:
   `\displaystyle \lim_(x \to 8^+) 10 - x`
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle \lim_(x \to 8^+) 10 - x = 10 - 8 = 2`

Step 3: Find Left-Hand Limit

1. Substitute in function [Limit]:
   `\displaystyle \lim_(x \to 8^-) x + 10`
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle \lim_(x \to 8^+) x + 10 = 8 + 10 = 18`

∴ since
`\displaystyle \lim_(x \to 8^+) f(x) \\eq \lim_(x \to 8^-) f(x)`,
`\displaystyle \lim_(x \to 8) f(x) = DNE`

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

Unit: Limits