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Question 2 (2 points)
Find the indicated limit, if it exists. (2 points)

1) 0
2) 8
3) 3
4) The limit does not exist.

Question 2 2 Points Find The Indicated Limit If It Exists 2 Points 1 0 2 8 3 3 4 The Limit Does Not Exist class=

Sagot :

Space

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

  1. Substitute in function [Left-Side Limit]:                                                       [tex]\displaystyle \lim_{x \to 5^-} 5 - x[/tex]
  2. 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

  1. Substitute in function [Right-Side Limit]:                                                     [tex]\displaystyle \lim_{x \to 5^+} x + 3[/tex]
  2. 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

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