Math Problem Statement
im asking to find what a is when the function is continuos on a line
Solution
To ensure that the function is continuous on the real line, we need to make sure that there is no discontinuity at , where the two pieces of the piecewise function meet.
The function is defined as:
\begin{cases} \frac{3 \sin x}{x}, & \text{if } x < 0 \\ a - 4x, & \text{if } x \geq 0 \end{cases}$$ ### Steps to Find $$a$$ for Continuity: 1. **Continuity at $$x = 0$$** requires that the limit of $$g(x)$$ as $$x$$ approaches 0 from the left (i.e., $$x \to 0^-$$) equals the limit as $$x$$ approaches 0 from the right (i.e., $$x \to 0^+$$), and also equals $$g(0)$$. 2. **Left-hand limit $$(x \to 0^-)$$**: $$\lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} \frac{3 \sin x}{x}$$ From the standard limit result: $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ Therefore, $$\lim_{x \to 0^-} g(x) = 3 \cdot 1 = 3$$ 3. **Right-hand limit $$(x \to 0^+)$$**: $$\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (a - 4x) = a$$ 4. **Set the left-hand limit equal to the right-hand limit for continuity**: $$3 = a$$ ### Conclusion: For the function $$g(x)$$ to be continuous at $$x = 0$$, the constant $$a$$ must be $$\mathbf{3}$$. Would you like to explore this solution further, or would you like to see a graphical representation? ### 5 Related Questions: 1. What happens if $$a$$ is not equal to 3? How does the function behave? 2. Can we also make the function differentiable at $$x = 0$$? 3. What are the implications of this continuity condition on the overall graph of the function? 4. How does the derivative of the function change at $$x = 0$$? 5. What other types of piecewise functions require similar continuity conditions? ### Tip: When determining the continuity of a piecewise function, always equate the limits from both sides at the boundary point where the function pieces meet.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Limit definition
Theorems
-
Suitable Grade Level
Advanced High School