Math Problem Statement
find these limits
find these limits
Solution
We have the following problem with the given function :
\begin{cases} x & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2 - x^2 & \text{if } 1 < x \leq 2 \\ x - 3 & \text{if } x > 2 \end{cases}$$ We are asked to find the following limits and values: ### (i) $$\lim_{x \to 1^-} g(x)$$ This is the left-hand limit as $$x$$ approaches 1 from the left. For $$x < 1$$, we know that $$g(x) = x$$, so: $$\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} x = 1$$ ### (ii) $$\lim_{x \to 1^+} g(x)$$ This is the right-hand limit as $$x$$ approaches 1 from the right. For $$1 < x \leq 2$$, $$g(x) = 2 - x^2$$. Therefore: $$\lim_{x \to 1^+} g(x) = 2 - 1^2 = 1$$ ### (iii) $$g(1)$$ From the definition of $$g(x)$$, when $$x = 1$$, we have: $$g(1) = 3$$ ### (iv) $$\lim_{x \to 2^-} g(x)$$ This is the left-hand limit as $$x$$ approaches 2 from the left. For $$1 < x \leq 2$$, $$g(x) = 2 - x^2$$, so: $$\lim_{x \to 2^-} g(x) = 2 - 2^2 = 2 - 4 = -2$$ ### (v) $$\lim_{x \to 2^+} g(x)$$ This is the right-hand limit as $$x$$ approaches 2 from the right. For $$x > 2$$, $$g(x) = x - 3$$, so: $$\lim_{x \to 2^+} g(x) = 2 - 3 = -1$$ ### (vi) $$\lim_{x \to 2} g(x)$$ Since $$\lim_{x \to 2^-} g(x) = -2$$ and $$\lim_{x \to 2^+} g(x) = -1$$, the two one-sided limits do not match. Therefore, the two-sided limit does not exist: $$\lim_{x \to 2} g(x) \text{ does not exist.}$$ ### Summary of Results: - (i) $$\lim_{x \to 1^-} g(x) = 1$$ - (ii) $$\lim_{x \to 1^+} g(x) = 1$$ - (iii) $$g(1) = 3$$ - (iv) $$\lim_{x \to 2^-} g(x) = -2$$ - (v) $$\lim_{x \to 2^+} g(x) = -1$$ - (vi) $$\lim_{x \to 2} g(x)$$ does not exist. Would you like further explanation on any of these limits? Here are some additional questions for you: 1. What conditions must be satisfied for a limit to exist at a point? 2. Why is it important to consider both left-hand and right-hand limits? 3. How does the piecewise nature of $$g(x)$$ affect the continuity of the function? 4. Can you find the limit of $$g(x)$$ as $$x \to \infty$$? 5. How would you define continuity at a point, based on the function $$g(x)$$? **Tip:** When evaluating limits, always consider the behavior of the function from both sides of the point to avoid missing any discontinuities.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Left-Hand and Right-Hand Limits
Formulas
Limit from the left: \( \lim_{x \to a^-} f(x) \)
Limit from the right: \( \lim_{x \to a^+} f(x) \)
Theorems
Limit Laws
Continuity of Piecewise Functions
Suitable Grade Level
Undergraduate Calculus or Advanced High School Calculus