Math Problem Statement
Find the limit of \( f(x) \) as \( x \) tends to 2 from the left if \( f(x) = \begin{cases} 1 + x^2 & \text{if } x < 2 \\ x^3 & \text{if } x \geq 2 \end{cases} \).
Solution
To solve this problem, we need to evaluate the limit of as approaches from the left, which means we need to use the expression defined for . The function is defined as:
1 + x^2 & \text{if } x < 2 \\ x^3 & \text{if } x \geq 2 \end{cases}$$ Since we are looking for the limit from the left side, we will use the expression $$f(x) = 1 + x^2$$ as $$x \to 2^-$$. ### Step-by-Step Solution 1. **Substitute $$x = 2$$ in the expression $$1 + x^2$$:** $$f(x) = 1 + x^2$$ 2. **Calculate $$f(2)$$ using this expression:** $$f(2) = 1 + (2)^2 = 1 + 4 = 5$$ Thus, the limit of $$f(x)$$ as $$x$$ approaches $$2$$ from the left is $$5$$. ### Answer The correct answer is **(a) 5**. Would you like further details on this calculation or have any questions? ### Related Questions 1. What is the limit of $$f(x)$$ as $$x$$ approaches $$2$$ from the right? 2. How does the definition of the function change for values of $$x$$ greater than or equal to 2? 3. Why is it necessary to approach from the left to use the expression $$1 + x^2$$? 4. What is the overall limit of $$f(x)$$ as $$x$$ approaches $$2$$ (if it exists)? 5. How would the limit change if the function were defined differently for $$x < 2$$? ### Tip When finding limits from one side, always ensure you're using the correct part of the function's piecewise definition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
Limit of a function as x approaches a point
Theorems
Limit definition for piecewise functions
Suitable Grade Level
Grades 11-12
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