Math Problem Statement
find the limits of f(x) as x approaches 2 from the right in this piecewise function, |x+3|, x<2 and x^2+1, x>2
Solution
The piecewise function is defined as:
\begin{cases} |x+3| & \text{if } x < 2 \\ x^2 + 1 & \text{if } x > 2 \end{cases}$$ We are interested in finding the limit of $$f(x)$$ as $$x$$ approaches 2 from the right, denoted as $$\lim_{x \to 2^+} f(x)$$. ### Step-by-Step Solution: For $$x > 2$$, the function is defined as $$f(x) = x^2 + 1$$. To find the right-hand limit as $$x \to 2$$: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x^2 + 1)$$ Substitute $$x = 2$$: $$\lim_{x \to 2^+} (x^2 + 1) = 2^2 + 1 = 4 + 1 = 5$$ Thus, the right-hand limit of $$f(x)$$ as $$x \to 2^+$$ is 5. ### Conclusion: $$\lim_{x \to 2^+} f(x) = 5$$ Do you want more details or have any questions? --- Here are 5 related questions you can explore: 1. What is the left-hand limit $$\lim_{x \to 2^-} f(x)$$? 2. Is $$f(x)$$ continuous at $$x = 2$$? 3. How do you evaluate limits for functions with absolute values? 4. What happens if $$x = 2$$ for the given piecewise function? 5. Can you graph this piecewise function and analyze its behavior around $$x = 2$$? **Tip:** Always check both the right-hand and left-hand limits to determine if a function is continuous at a point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Right-hand Limits
Formulas
Limit formula for right-hand limits: lim_{x → c^+} f(x)
Piecewise function definition
Theorems
Limit Laws
Suitable Grade Level
Grades 11-12