Math Problem Statement
Why is the following function discontinuous at x = 0?
Solution
To determine why the given function is discontinuous at , let us analyze the problem in steps. The function is defined as follows:
\begin{cases} e^x, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$$ ### Steps to Analyze Discontinuity at $$x = 0$$ 1. **Evaluate $$f(0)$$:** Since $$x \geq 0$$ applies when $$x = 0$$, $$f(0) = 0^2 = 0$$. Thus, $$f(0)$$ exists and equals 0. 2. **Evaluate the left-hand limit ($$\lim_{x \to 0^-} f(x)$$):** For $$x < 0$$, $$f(x) = e^x$$. As $$x \to 0^-$$, $$e^x \to e^0 = 1$$. So, $$\lim_{x \to 0^-} f(x) = 1$$. 3. **Evaluate the right-hand limit ($$\lim_{x \to 0^+} f(x)$$):** For $$x \geq 0$$, $$f(x) = x^2$$. As $$x \to 0^+$$, $$x^2 \to 0^2 = 0$$. So, $$\lim_{x \to 0^+} f(x) = 0$$. 4. **Compare the left-hand and right-hand limits:** $$\lim_{x \to 0^-} f(x) = 1 \neq 0 = \lim_{x \to 0^+} f(x)$$ Since the left-hand limit and right-hand limit are not equal, the two-sided limit $$\lim_{x \to 0} f(x)$$ does not exist. 5. **Conclusion:** The function is discontinuous at $$x = 0$$ because the limit $$\lim_{x \to 0} f(x)$$ does not exist, even though $$f(0)$$ exists. Therefore, the correct answer is **(b): $$\lim_{x \to 0} f(x)$$ does not exist (or is infinite).** --- ### Do you have any questions, or would you like further clarification? --- ### Related Questions: 1. What does it mean for a function to be continuous at a point? 2. How are left-hand and right-hand limits used to evaluate continuity? 3. Can a function be discontinuous even if $$f(0)$$ and $$\lim_{x \to 0} f(x)$$ are finite but unequal? 4. What are the common types of discontinuities in functions? 5. How does the exponential function $$e^x$$ behave as $$x \to 0^-$$? --- ### Tip: To check for continuity at a point, always evaluate $$f(c)$$, $$\lim_{x \to c^-} f(x)$$, and $$\lim_{x \to c^+} f(x)$$ systematically. If all match, the function is continuous at $$c$$.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 evaluation \( \lim_{x \to c} f(x) \)
Left-hand and right-hand limits
Theorems
Definition of continuity at a point
Suitable Grade Level
Undergraduate Mathematics / Advanced High School
Related Recommendation
Continuity of a Piecewise Function at x = 0 and x = 4
Determine Continuity and Type of Discontinuity for a Piecewise Function
Determine the Discontinuity Points of f(x) and Their Nature
Continuity and Graph of y = 1 / (2 - e^(x^2)) at x1 = 0 and x2 = 2
Piecewise Function Continuity - Graphing and Limit Evaluation