Question

Find the indicated limit, if it exists.

The limit is approaching 5.

Possible Options:
a) 0
b) 8
c) 3
d) The limit does not exist​

Answer 1

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)`

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) \\eq \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