Math Problem Statement
Determine the limit by sketching an appropriate graph:
\(
\lim_{x \to 4^-} f(x)
\)
where \( f(x) \) is defined as:
\[
f(x) =
\begin{cases}
\sqrt{1 - x^2} & \text{for } 0 \leq x < 1 \\
1 & \text{for } 1 \leq x < 4 \\
4 & \text{for } x = 4
\end{cases}
\]
Options:
- 0
- 1
- 4
- Does not exist
Solution
To analyze the given limit, let's break it down step by step.
We are asked to evaluate:
Where the function is piecewise defined as:
\begin{cases} \sqrt{1 - x^2} & \text{for } 0 \leq x < 1 \\ 1 & \text{for } 1 \leq x < 4 \\ 4 & \text{for } x = 4 \end{cases}$$ Now, let's analyze what happens as $$x \to 4^-$$ (from the left side): - For $$1 \leq x < 4$$, the function $$f(x) = 1$$. This means that as $$x$$ approaches 4 from values less than 4 but greater than or equal to 1, $$f(x)$$ is constantly equal to 1. Therefore: $$\lim_{x \to 4^-} f(x) = 1$$ The correct answer is: $$\boxed{1}$$ Would you like further details on how the piecewise function behaves graphically or any other aspects of the solution? Here are five related questions to explore: 1. What is the limit of $$f(x)$$ as $$x \to 4^+$$? 2. How does the function behave as $$x \to 0$$? 3. What would the graph of $$\sqrt{1 - x^2}$$ look like for $$0 \leq x < 1$$? 4. What is the importance of continuity when dealing with piecewise functions? 5. How would the limit change if the piecewise function was different at $$x = 4$$? **Tip**: When analyzing limits, always pay attention to whether you're approaching from the left (denoted $$x \to c^-$$) or from the right (denoted $$x \to c^+$$), especially with piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Approaching from the Left
Formulas
Limit Definition
Piecewise Function Evaluation
Theorems
Limit Laws
Piecewise Function Behavior
Suitable Grade Level
Grades 10-12