Math Problem Statement
also calculate Left hand limit,right hand limit and one sided limit
also calculate Left hand limit,right hand limit and one sided limit
Solution
To calculate the left-hand limit, right-hand limit, and two-sided limit for the given piecewise function , let's analyze it step by step:
Given:
\begin{cases} x - 1 & \text{if } x \leq 3, \\ 3x - 7 & \text{if } x > 3. \end{cases}$$ We need to find: 1. $$\lim_{x \to 3^-} f(x)$$ (left-hand limit) 2. $$\lim_{x \to 3^+} f(x)$$ (right-hand limit) 3. $$\lim_{x \to 3} f(x)$$ (two-sided limit) --- ### Step 1: Left-hand limit ($$\lim_{x \to 3^-} f(x)$$) The **left-hand limit** corresponds to values of $$x$$ approaching $$3$$ from the left side ($$x \leq 3$$). For $$x \leq 3$$, the function is: $$f(x) = x - 1.$$ Substitute $$x = 3$$ into $$x - 1$$: $$\lim_{x \to 3^-} f(x) = 3 - 1 = 2.$$ --- ### Step 2: Right-hand limit ($$\lim_{x \to 3^+} f(x)$$) The **right-hand limit** corresponds to values of $$x$$ approaching $$3$$ from the right side ($$x > 3$$). For $$x > 3$$, the function is: $$f(x) = 3x - 7.$$ Substitute $$x = 3$$ into $$3x - 7$$: $$\lim_{x \to 3^+} f(x) = 3(3) - 7 = 9 - 7 = 2.$$ --- ### Step 3: Two-sided limit ($$\lim_{x \to 3} f(x)$$) The **two-sided limit** exists if and only if the left-hand limit and right-hand limit are equal. From Steps 1 and 2: $$\lim_{x \to 3^-} f(x) = 2 \quad \text{and} \quad \lim_{x \to 3^+} f(x) = 2.$$ Since both are equal, the two-sided limit is: $$\lim_{x \to 3} f(x) = 2.$$ --- ### Final Answers: 1. $$\lim_{x \to 3^-} f(x) = 2$$ (Left-hand limit) 2. $$\lim_{x \to 3^+} f(x) = 2$$ (Right-hand limit) 3. $$\lim_{x \to 3} f(x) = 2$$ (Two-sided limit) --- ### Conclusion: The function $$f(x)$$ is continuous at $$x = 3$$, as the left-hand limit, right-hand limit, and the two-sided limit all exist and are equal to $$2$$. --- ### Follow-Up Questions: 1. What does it mean for a function to be **continuous** at a point? 2. What happens if the left-hand limit and right-hand limit are not equal? 3. How would the limit change if $$f(x)$$ had a discontinuity at $$x = 3$$? 4. Can you plot the graph of this function to visualize the limits? 5. How can this method be applied to more complex piecewise functions? **Tip**: Always check the continuity of a piecewise function at the boundary points by verifying the left-hand limit, right-hand limit, and the function value.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
One-Sided Limits
Formulas
Left-hand limit: lim_{x → c^-} f(x)
Right-hand limit: lim_{x → c^+} f(x)
Two-sided limit: lim_{x → c} f(x)
Theorems
Definition of Limits
Continuity at a Point
Suitable Grade Level
Grades 11-12