Math Problem Statement
Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let f(x) = { -2x + 3 if x ≤ 1; -7x + 8 if x > 1 }. Find lim_{x → 1} f(x).
Solution
The problem requires finding the two-sided limit , given the piecewise function:
\begin{cases} -2x + 3 & \text{if } x \leq 1, \\ -7x + 8 & \text{if } x > 1. \end{cases}$$ ### Step 1: Evaluate the Left-Hand Limit ($$\lim_{x \to 1^-} f(x)$$) For $$x \leq 1$$, $$f(x) = -2x + 3$$. Substituting $$x = 1$$ into $$-2x + 3$$: $$\lim_{x \to 1^-} f(x) = -2(1) + 3 = -2 + 3 = 1.$$ ### Step 2: Evaluate the Right-Hand Limit ($$\lim_{x \to 1^+} f(x)$$) For $$x > 1$$, $$f(x) = -7x + 8$$. Substituting $$x = 1$$ into $$-7x + 8$$: $$\lim_{x \to 1^+} f(x) = -7(1) + 8 = -7 + 8 = 1.$$ ### Step 3: Compare the Left-Hand and Right-Hand Limits Since $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = 1$$, the two-sided limit exists, and its value is: $$\lim_{x \to 1} f(x) = 1.$$ ### Final Answer: **A. 1** --- Would you like further details or have any questions? ### Expanding Questions: 1. What happens if the left-hand and right-hand limits differ at a point? 2. Can you describe the conditions for a limit to exist? 3. How would this problem change if $$x = 1$$ was excluded from the function definition? 4. Why is continuity not necessary for limits to exist? 5. How can we visualize this piecewise function graphically? ### Tip: When working with piecewise functions, always evaluate the limit from both sides of the critical point to ensure consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Left-hand and Right-hand Limits
Formulas
Limit formula: lim_{x → c} f(x) = L if lim_{x → c^-} f(x) = lim_{x → c^+} f(x) = L
Theorems
Definition of Two-sided Limits
Suitable Grade Level
Grades 11-12