Math Problem Statement
Find the limit of what is in the image
Find the limit of what is in the image
Solution
The limit expression in the image is:
\begin{cases} x, & \text{if } x < 2 \\ x + 1, & \text{if } x \geq 2 \end{cases}$$ ### Step-by-step solution: We need to consider the left-hand limit ($$x \to 2^{-}$$) and the right-hand limit ($$x \to 2^{+}$$). 1. **Left-hand limit** ($$x \to 2^{-}$$): - For $$x < 2$$, $$f(x) = x$$. - So, as $$x \to 2^{-}$$, $$f(x) \to 2$$. Therefore, $$\lim_{{x \to 2^{-}}} f(x) = 2.$$ 2. **Right-hand limit** ($$x \to 2^{+}$$): - For $$x \geq 2$$, $$f(x) = x + 1$$. - So, as $$x \to 2^{+}$$, $$f(x) = 2 + 1 = 3$$. Therefore, $$\lim_{{x \to 2^{+}}} f(x) = 3.$$ ### Conclusion: Since the left-hand limit is 2 and the right-hand limit is 3, and they are not equal, the limit **does not exist** at $$x = 2$$. Would you like any more details on this solution or further clarification? Here are five related questions to think about: 1. What conditions need to be met for a limit to exist at a point? 2. How does one compute one-sided limits in piecewise functions? 3. What is the significance of discontinuities in a function's limit? 4. Can the value of the function at the point $$x = 2$$ affect the existence of the limit? 5. How do limits relate to the continuity of a function at a point? **Tip:** When solving limit problems for piecewise functions, always check the behavior from both sides of the point of interest to determine if the limit exists!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
One-Sided Limits
Formulas
Limit definition
Piecewise function limit calculation
Theorems
Limit existence theorem
Left-hand limit vs right-hand limit
Suitable Grade Level
Grades 11-12 (Pre-Calculus/Calculus)