Math Problem Statement
f(x) = { x^2 + 1 if x < 1, (x - 3)^2 if x >= 1}. Find the following limits: lim(x→1^-)f(x), lim(x→1^+)f(x).
Solution
The function is given as:
\begin{cases} x^2 + 1 & \text{if } x < 1 \\ (x - 3)^2 & \text{if } x \geq 1 \end{cases}$$ ### (a) Find the following limits: 1. $$\lim\limits_{x \to 1^-} f(x)$$ This is the limit as $$x$$ approaches 1 from the left. For $$x < 1$$, the function is given by $$f(x) = x^2 + 1$$. So we compute: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2$$ 2. $$\lim\limits_{x \to 1^+} f(x)$$ This is the limit as $$x$$ approaches 1 from the right. For $$x \geq 1$$, the function is given by $$f(x) = (x - 3)^2$$. So we compute: $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x - 3)^2 = (1 - 3)^2 = (-2)^2 = 4$$ ### Conclusion: Since $$\lim_{x \to 1^-} f(x) = 2$$ and $$\lim_{x \to 1^+} f(x) = 4$$, the two one-sided limits are not equal. Therefore, the overall limit $$\lim_{x \to 1} f(x)$$ does not exist (DNE). --- Would you like further details or explanations? Here are five related questions to expand on this topic: 1. What is the definition of a one-sided limit? 2. How does the behavior of a piecewise function affect the existence of a limit at a boundary point? 3. Can a limit exist if the one-sided limits are unequal? Why or why not? 4. How can you graphically interpret one-sided limits? 5. What is the importance of continuity in relation to limits? **Tip:** When dealing with piecewise functions, always check both one-sided limits at the point of interest to determine if the overall limit exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
One-sided Limits
Formulas
f(x) = x^2 + 1
f(x) = (x - 3)^2
Theorems
Definition of a Limit
One-sided Limit Theorem
Suitable Grade Level
Grades 11-12