Answer:
4) 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 [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Step-by-step explanation:
*Note:
For a limit to exist, the right-side and left-side limits must be equal to each other.
Step 1: Define
Identify
[tex]\displaystyle f(x) = \left\{\begin{array}{ccc}5 - x ,\ x < 5\\8 ,\ x = 5\\x + 3 ,\ x > 5\end{array}[/tex]
Step 2: Find Left-Side Limit
- Substitute in function [Left-Side Limit]: [tex]\displaystyle \lim_{x \to 5^-} 5 - x[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 5^-} 5 - x = 5- 5 = 0[/tex]
Step 2: Find Left-Side Limit
- Substitute in function [Right-Side Limit]: [tex]\displaystyle \lim_{x \to 5^+} x + 3[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 5^+} x + 3 = 5 + 3 = 8[/tex]
∴ since [tex]\displaystyle \lim_{x \to c^+} f(x) \neq \lim_{x \to c^-} f(x)[/tex] , [tex]\displaystyle \lim_{x \to 5} f(x) = \text{DNE}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
