Math Problem Statement
Determine if the function f(x) = { x + e^x, x < 0; x^2, x >= 0 } is continuous or discontinuous at the points x = 0 and x = 4.
Solution
We are given the piecewise function:
\begin{cases} x + e^x, & x < 0 \\ x^2, & x \geq 0 \end{cases}$$ We need to determine if this function is continuous or discontinuous at the points $$x = 0$$ and $$x = 4$$. ### 1. Continuity at $$x = 0$$ A function is continuous at $$x = 0$$ if: - $$f(0)$$ is defined. - $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x)$$. - $$\lim_{x \to 0} f(x) = f(0)$$. #### Checking these conditions: - $$f(0) = 0^2 = 0$$ (using the second case of the piecewise function). - The left-hand limit $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + e^x)$$. Since $$e^x \to 1$$ as $$x \to 0$$, we get: $$\lim_{x \to 0^-} (x + e^x) = 0 + 1 = 1.$$ - The right-hand limit $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0^2 = 0$$. Since $$\lim_{x \to 0^-} f(x) = 1$$ and $$\lim_{x \to 0^+} f(x) = 0$$, the limits are not equal. Therefore, the function is **discontinuous** at $$x = 0$$. ### 2. Continuity at $$x = 4$$ Here, $$x = 4$$ falls within the second piece of the function, so we just check the limit at $$x = 4$$. The function is defined by $$f(x) = x^2$$ when $$x \geq 0$$, so: - $$f(4) = 4^2 = 16$$. - The limit $$\lim_{x \to 4} f(x) = \lim_{x \to 4} x^2 = 16$$. Since the limit exists and equals $$f(4)$$, the function is **continuous** at $$x = 4$$. ### Conclusion: - The function is **discontinuous** at $$x = 0$$. - The function is **continuous** at $$x = 4$$. Would you like more details or further clarification? Here are 5 related questions for further exploration: 1. How do we determine the differentiability of a piecewise function at a given point? 2. Can a function be continuous but not differentiable at a point? If so, what are the conditions? 3. What happens to the limits if we modify the function for $$x < 0$$ to be $$x^2 + e^x$$? 4. How would the continuity of $$f(x)$$ change if we defined a different value for $$f(0)$$? 5. How can we visualize the discontinuity at $$x = 0$$ on a graph? **Tip:** Always check both the left-hand and right-hand limits separately when analyzing the continuity of piecewise functions at boundary points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Limit definition for continuity: \lim_{x \to c} f(x) = f(c)
Theorems
Continuity Theorem
Piecewise Function Limit
Suitable Grade Level
Grades 10-12