Answer:
The limit does not exist
General Formulas and Concepts:
Calculus
Limits
- Right-Side Limit: [tex]\displaystyle \lim_{x \to c^+} f(x)[/tex]
- Left-Side Limit: [tex]\displaystyle \lim_{x \to c^-} f(x)[/tex]
Limit Rule [Constant]: [tex]\displaystyle \lim_{x \to c} b = b[/tex]
Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Limit Property [Addition/Subtraction]: [tex]\displaystyle \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)[/tex]
Step-by-step explanation:
*Note:
In order for the limit to exist, the right-hand and left-hand limits must equal each other
Step 1: Define
Identify
[tex]\displaystyle f(x) = \left \{ {{x + 10,\ x < 8} \atop {10 - x,\ x \geq 8}} \right.[/tex]
Step 2: Find Right-Hand Limit
- Substitute in function [Limit]: [tex]\displaystyle \lim_{x \to 8^+} 10 - x[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 8^+} 10 - x = 10 - 8 = 2[/tex]
Step 3: Find Left-Hand Limit
- Substitute in function [Limit]: [tex]\displaystyle \lim_{x \to 8^-} x + 10[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 8^+} x + 10 = 8 + 10 = 18[/tex]
∴ since [tex]\displaystyle \lim_{x \to 8^+} f(x) \neq \lim_{x \to 8^-} f(x)[/tex], [tex]\displaystyle \lim_{x \to 8} f(x) = DNE[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
