Math Problem Statement
Determine the limit of the piecewise function f(x) as x approaches 1.
Solution
The problem asks to determine the limit of the piecewise function as .
The function is defined as:
\begin{cases} x + 1 & \text{if } x < 1 \\ -x^2 + 4x - 1 & \text{if } x \geq 1 \end{cases}$$ To solve the limit $$\lim_{x \to 1} f(x)$$, we need to check the left-hand limit and the right-hand limit separately, then compare them. ### Left-hand limit ($$x \to 1^{-}$$): When $$x \to 1$$ from the left ($$x < 1$$), we use the part of the function $$f(x) = x + 1$$. $$\lim_{x \to 1^{-}} f(x) = \lim_{x \to 1^{-}} (x + 1) = 1 + 1 = 2$$ ### Right-hand limit ($$x \to 1^{+}$$): When $$x \to 1$$ from the right ($$x \geq 1$$), we use the part of the function $$f(x) = -x^2 + 4x - 1$$. $$\lim_{x \to 1^{+}} f(x) = \lim_{x \to 1^{+}} (-x^2 + 4x - 1) = -(1)^2 + 4(1) - 1 = -1 + 4 - 1 = 2$$ ### Conclusion: Since both the left-hand limit and right-hand limit are equal to 2, the overall limit exists and is: $$\lim_{x \to 1} f(x) = 2$$ Would you like more details or have any further questions? Here are 5 related questions to expand your understanding: 1. What is the general process to solve limits for piecewise functions? 2. How do you determine continuity at $$x = 1$$ for this function? 3. What happens if the left-hand and right-hand limits do not agree? 4. How would the limit change if the functions on either side of $$x = 1$$ were different? 5. Can you find the derivative of $$f(x)$$ at $$x = 1$$ using limits? **Tip:** Always check both the left-hand and right-hand limits when dealing with piecewise functions. The function's limit exists at a point if and only if both limits are equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Left-hand limit: \lim_{x \to 1^-} f(x)
Right-hand limit: \lim_{x \to 1^+} f(x)
Theorems
Definition of a limit
Continuity at a point
Suitable Grade Level
Grades 11-12 or Early College