Question

Find the indicated limit, if it exists.

Answer 1

Answer:

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

Limit Property [Addition/Subtraction]:                                                                   $\displaystyle \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$

Step-by-step explanation:

*Note:

In order for a limit to exist, the right-side and left-side limits must equal 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 Right-Side Limit

1. Substitute in function [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 3: Find Left-Side Limit

1. Substitute in function [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  $\displaystyle \lim_{x \to 5^+} f(x) \neq \lim_{x \to 5^-} f(x)$  , then  $\displaystyle \lim_{x \to 5} f(x) = DNE$

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

Unit:  Limits