Math Problem Statement
show that the function FX sin x / X ,x not equals to 0 and 0 ,x = 0 is not continuous at X = 0
Solution
To analyze the continuity of the given function , let's define it formally:
\begin{cases} \frac{\sin x}{x}, & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases}$$ ### Step 1: Continuity Condition A function $$f(x)$$ is continuous at $$x = 0$$ if: $$\lim_{x \to 0} f(x) = f(0).$$ Here, $$f(0) = 0$$ (given). Therefore, we need to check if: $$\lim_{x \to 0} f(x) = 0.$$ ### Step 2: Compute $$\lim_{x \to 0} f(x)$$ For $$x \neq 0$$, $$f(x) = \frac{\sin x}{x}$$. Thus: $$\lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin x}{x}.$$ From standard calculus results, we know: $$\lim_{x \to 0} \frac{\sin x}{x} = 1.$$ ### Step 3: Compare the Limit and $$f(0)$$ The left-hand limit is $$1$$, but $$f(0) = 0$$. Since: $$\lim_{x \to 0} f(x) \neq f(0),$$ the function $$f(x)$$ is **not continuous at $$x = 0$$**. --- ### Conclusion The function $$f(x)$$ is not continuous at $$x = 0$$ because the limit of $$f(x)$$ as $$x \to 0$$ does not equal the value of the function at $$x = 0$$. --- ### Would you like more details or examples about continuity? ### Here are 5 related questions you might explore: 1. What is the formal definition of continuity for a piecewise function? 2. How can $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ be proven rigorously? 3. Can a discontinuous function be made continuous by redefining it? 4. What types of discontinuities exist in functions (e.g., removable, jump, infinite)? 5. How does the function $$\frac{\sin x}{x}$$ behave near $$x = 0$$? **Tip:** When checking continuity, always verify that the limit equals the function's value at the point of interest.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Limits
Piecewise functions
Formulas
Definition of continuity: lim(x → c) f(x) = f(c)
Standard limit: lim(x → 0) (sin(x)/x) = 1
Theorems
Limit of sin(x)/x as x approaches 0
Suitable Grade Level
Grades 11-12
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