Math Problem Statement
Discuss the continuity of: f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩cos(x)x<−π|cos(x)|−π≤x≤π2cos2(x)x>π2⎫⎪ ⎪⎬⎪ ⎪⎭
Solution
To discuss the continuity of the piecewise function
\begin{cases} \cos(x) & \text{if } x < -\pi \\ |\cos(x)| & \text{if } -\pi \leq x \leq \pi \\ 2\cos^2(x) & \text{if } x > \pi \end{cases}$$ we need to examine its continuity across its domain, particularly at the points where the pieces of the function change, namely $$x = -\pi$$ and $$x = \pi$$. ### 1. Continuity at $$x = -\pi$$ - **Left-hand limit:** Consider $$\lim_{x \to -\pi^-} f(x)$$. Since $$f(x) = \cos(x)$$ for $$x < -\pi$$, we calculate: $$\lim_{x \to -\pi^-} f(x) = \cos(-\pi) = -1$$ - **Right-hand limit:** Consider $$\lim_{x \to -\pi^+} f(x)$$. Since $$f(x) = |\cos(x)|$$ for $$-\pi \leq x \leq \pi$$, we calculate: $$\lim_{x \to -\pi^+} f(x) = |\cos(-\pi)| = 1$$ - **Value of the function at $$x = -\pi$$:** Since $$f(x) = |\cos(x)|$$ for $$-\pi \leq x \leq \pi$$, we have: $$f(-\pi) = |\cos(-\pi)| = 1$$ Since the left-hand limit ($$-1$$) and right-hand limit ($$1$$) at $$x = -\pi$$ are not equal, $$f(x)$$ is **not continuous** at $$x = -\pi$$. ### 2. Continuity at $$x = \pi$$ - **Left-hand limit:** Consider $$\lim_{x \to \pi^-} f(x)$$. Since $$f(x) = |\cos(x)|$$ for $$-\pi \leq x \leq \pi$$, we calculate: $$\lim_{x \to \pi^-} f(x) = |\cos(\pi)| = 1$$ - **Right-hand limit:** Consider $$\lim_{x \to \pi^+} f(x)$$. Since $$f(x) = 2\cos^2(x)$$ for $$x > \pi$$, we calculate: $$\lim_{x \to \pi^+} f(x) = 2\cos^2(\pi) = 2(-1)^2 = 2$$ - **Value of the function at $$x = \pi$$:** Since $$f(x) = |\cos(x)|$$ for $$-\pi \leq x \leq \pi$$, we have: $$f(\pi) = |\cos(\pi)| = 1$$ Since the left-hand limit ($$1$$) and right-hand limit ($$2$$) at $$x = \pi$$ are not equal, $$f(x)$$ is **not continuous** at $$x = \pi$$. ### Conclusion The function $$f(x)$$ is **not continuous** at $$x = -\pi$$ and $$x = \pi$$, due to mismatched left-hand and right-hand limits at these points. The function is continuous elsewhere. Would you like further details or have any questions? Here are some related questions: 1. How do we define continuity for a piecewise function? 2. What are the implications of discontinuity in a function? 3. Can a function be continuous everywhere but not differentiable at some points? 4. How would the function $$f(x)$$ change if we adjusted the piecewise conditions? 5. What are some examples of real-world phenomena modeled by discontinuous functions? **Tip:** When checking for continuity at a point in a piecewise function, always compare the left-hand limit, right-hand limit, and the function's value at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Continuity
Limits
Formulas
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Theorems
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Suitable Grade Level
College Level
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