Question

Question 2 (2 points)

Answer 1

Answer:

4) 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 [Variable Direct Substitution]:                                                             $\displaystyle \lim_{x \to c} x = c$

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

$\displaystyle f(x) = \left\{\begin{array}{ccc}5 - x ,\ x < 5\\8 ,\ x = 5\\x + 3 ,\ x > 5\end{array}$

Step 2: Find Left-Side Limit

1. Substitute in function [Left-Side Limit]:                                                       $\displaystyle \lim_{x \to 5^-} 5 - x$
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:                          $\displaystyle \lim_{x \to 5^-} 5 - x = 5- 5 = 0$

Step 2: Find Left-Side Limit

1. Substitute in function [Right-Side Limit]:                                                     $\displaystyle \lim_{x \to 5^+} x + 3$
2. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           $\displaystyle \lim_{x \to 5^+} x + 3 = 5 + 3 = 8$

∴ since  $\displaystyle \lim_{x \to c^+} f(x) \neq \lim_{x \to c^-} f(x)$  ,  $\displaystyle \lim_{x \to 5} f(x) = \text{DNE}$

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

Unit: Limits