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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![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89524ea7-6be4-4cd1-8a76-e790f4380b75%2F8dfc3add-7469-4ecd-b9aa-80845cd33342%2Faaof3ao.png&w=3840&q=75)
Transcribed Image Text:### 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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