Math Problem Statement
Find the limits as x approaches 1 from the left and right side for f(x) = (x^2 - 3, 0≤x<1), (-1, x=1), (3x - 5, 1<x)
Solution
We are tasked with finding the left-hand limit () and the right-hand limit () for the given piecewise function:
\begin{cases} x^2 - 3, & 0 \leq x < 1 \\ -1, & x = 1 \\ 3x - 5, & x > 1 \end{cases}$$ ### Step 1: Evaluate $$\lim_{x \to 1^-} f(x)$$ As $$x \to 1^-$$, the function corresponds to the first piece $$f(x) = x^2 - 3$$. Substituting $$x = 1$$ into $$x^2 - 3$$, we get: $$\lim_{x \to 1^-} f(x) = 1^2 - 3 = -2.$$ ### Step 2: Evaluate $$\lim_{x \to 1^+} f(x)$$ As $$x \to 1^+$$, the function corresponds to the third piece $$f(x) = 3x - 5$$. Substituting $$x = 1$$ into $$3x - 5$$, we get: $$\lim_{x \to 1^+} f(x) = 3(1) - 5 = -2.$$ ### Step 3: Compare the left-hand and right-hand limits Since both the left-hand and right-hand limits are equal ($$-2$$), the two-sided limit exists and is given by: $$\lim_{x \to 1} f(x) = -2.$$ ### Step 4: Note about the value of $$f(x)$$ at $$x = 1$$ The function is explicitly defined as $$f(1) = -1$$, which is different from the limit as $$x \to 1$$. Therefore, $$f(x)$$ has a discontinuity at $$x = 1$$. --- ### Final Answer: $$\lim_{x \to 1^-} f(x) = -2, \quad \lim_{x \to 1^+} f(x) = -2.$$ Would you like a detailed graph or further clarification? 😊 --- ### Related Questions: 1. What are the different types of discontinuities in piecewise functions? 2. How do you determine if a function is continuous at a point? 3. Can a two-sided limit exist if the left and right limits differ? 4. How do you graph piecewise functions? 5. What is the difference between a limit and the value of a function at a point? ### Tip: Always check each piece of a piecewise function carefully when evaluating limits, as the behavior of the function may change at the boundary points.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 -> c^-} f(x)
Right-hand limit: lim_{x -> c^+} f(x)
Theorems
Limit Theorem
Continuity Criterion
Suitable Grade Level
Grades 10-12