### Understanding Limits**Recall this Fact:** For a limit to exist, it means the limit from the left and from the right must exist and be equal:\[ \lim_{{x \to a}} f(x) = L \iff \lim_{{x \to a^{-}}} f(x) = L = \lim_{{x \to a^{+}}} f(x). \]### Exercise To Try 9Using the diagrams below, determine if either limit exists.#### a) $\lim_{{x \to 0}} |x| = \_\_\_\_\_\_\_**Explanation of the diagram:**- The graph on the left represents \( y = |x| $.- The x-axis ranges from -3 to 3, and the y-axis ranges from 0 to 3.- The graph is a V-shaped curve converging at the origin (0,0).#### b) $\lim_{{x \to 0}} \frac{|x|}{x} = \_\_\_\_\_\_\_**Explanation of the diagram:**- The graph on the right represents \( y = \frac{|x|}{x} $.- The x-axis ranges from -3 to 3, and the y-axis ranges from -2 to 2.- The graph has two horizontal lines: one at $ y = 1 $ for $ x > 0 $ and another at $ y = -1 $ for $ x < 0 $.- There is a gap in the graph at the origin (x=0), represented with open circles, indicating that the function is not defined at that point.Please use the provided graphs to evaluate the limits and determine if they exist.

Answered: Image /qna-images/answer/8dfc3add-7469-4ecd-b9aa-80845cd33342.jpg

Recall this Fact: For a limit to exist means the limit from the left and from the right must exist and be equal: lim f(x) = L→ lim f(x) = L = lim f(x) Exercise To Try 9 Using the diagrams below, determine if either limit erists. a) lim Ja| b) lim X→0 y=x| y = |x\/x 1.5 2.5 2 0.5 >1.5 -0.5 1 0.5 -1.5 -3 -2 -1 2 3

Recall this Fact: For a limit to exist means the limit from the left and from the right must exist and be equal: lim f(x) = L→ lim f(x) = L = lim f(x) Exercise To Try 9 Using the diagrams below, determine if either limit erists. a) lim Ja| b) lim X→0 y=x| y = |x\/x 1.5 2.5 2 0.5 >1.5 -0.5 1 0.5 -1.5 -3 -2 -1 2 3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

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.

Question

### Understanding Limits

**Recall this Fact:** For a limit to exist, it means the limit from the left and from the right must exist and be equal:
\[ \lim_{{x \to a}} f(x) = L \iff \lim_{{x \to a^{-}}} f(x) = L = \lim_{{x \to a^{+}}} f(x). \]

### Exercise To Try 9

Using the diagrams below, determine if either limit exists.

#### a) $\lim_{{x \to 0}} |x| = \_\_\_\_\_\_\_

**Explanation of the diagram:**
- The graph on the left represents \( y = |x| $.
- The x-axis ranges from -3 to 3, and the y-axis ranges from 0 to 3.
- The graph is a V-shaped curve converging at the origin (0,0).

#### b) $\lim_{{x \to 0}} \frac{|x|}{x} = \_\_\_\_\_\_\_

**Explanation of the diagram:**
- The graph on the right represents \( y = \frac{|x|}{x} $.
- The x-axis ranges from -3 to 3, and the y-axis ranges from -2 to 2.
- The graph has two horizontal lines: one at $ y = 1 $ for $ x > 0 $ and another at $ y = -1 $ for $ x < 0 $.
- There is a gap in the graph at the origin (x=0), represented with open circles, indicating that the function is not defined at that point.

Please use the provided graphs to evaluate the limits and determine if they exist.

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