### Limit Evaluation Problem**Problem 4:** Determine the following limit:\[ \lim_{{x \to 2}} \frac{1}{{x - 2}} \]**Diagram Explanation:**The diagram provided is a graph of the function $ f(x) = \frac{1}{{x - 2}} $. The graph shows a vertical asymptote at $ x = 2 $.- As $ x $ approaches $ 2 $ from the left ($ x \to 2^{-} $), the function value $ f(x) $ decreases without bound, heading towards negative infinity. - Conversely, as $ x $ approaches $ 2 $ from the right ($ x \to 2^{+} $), the function value $ f(x) $ increases without bound, heading towards positive infinity.This behavior indicates that the limit of $ \frac{1}{{x - 2}} $ as $ x $ approaches 2 does not approach a single finite value.**Options:**- a) -2- b) 0- c) -4- d) 2- e) Does not exist/undetermined### Answer:The correct choice is:- **e) Does not exist/undetermined**The limit does not exist because the function approaches negative infinity from the left and positive infinity from the right as $ x $ approaches 2.

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Answered: 4. Determine the following limit * 1…

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4. Determine the following limit * 1 lim * → 2 *- 2 3- -6- a) -2 b) 0 -4 d) 2 e) Does not exist/undetermined

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter9: Polynomial And Rational Functions

Section9.6: More On Graphing Rational Functions

Problem 28PS

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

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### Limit Evaluation Problem

**Problem 4:** Determine the following limit:
\[ \lim_{{x \to 2}} \frac{1}{{x - 2}} \]

**Diagram Explanation:**
The diagram provided is a graph of the function $ f(x) = \frac{1}{{x - 2}} $. The graph shows a vertical asymptote at $ x = 2 $.

- As $ x $ approaches $ 2 $ from the left ($ x \to 2^{-} $), the function value $ f(x) $ decreases without bound, heading towards negative infinity.

- Conversely, as $ x $ approaches $ 2 $ from the right ($ x \to 2^{+} $), the function value $ f(x) $ increases without bound, heading towards positive infinity.

This behavior indicates that the limit of $ \frac{1}{{x - 2}} $ as $ x $ approaches 2 does not approach a single finite value.

**Options:**

- a) -2
- b) 0
- c) -4
- d) 2
- e) Does not exist/undetermined

### Answer:
The correct choice is:
- **e) Does not exist/undetermined**

The limit does not exist because the function approaches negative infinity from the left and positive infinity from the right as $ x $ approaches 2.

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