Math Problem Statement
Given the following piecewise function, evaluate limx→1−f(x).
f(x)=⎧⎨⎩−2x2−2xx<−23x−2−2<x<1x2+3x>1
Solution
To evaluate the left-hand limit for the given piecewise function, we need to find which piece of the function corresponds to the case when approaches 1 from the left side.
The given piecewise function is:
\begin{cases} -2x^2 - 2x & \text{for } x < -2 \\ 3x - 2 & \text{for } -2 < x < 1 \\ x^2 + 3 & \text{for } x > 1 \end{cases}$$ Since we're interested in $$\lim_{x \to 1^-} f(x)$$, the relevant expression for $$f(x)$$ is the piece defined for $$-2 < x < 1$$, which is: $$f(x) = 3x - 2.$$ Now, take the limit as $$x$$ approaches 1 from the left: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (3x - 2).$$ Substitute $$x = 1$$: $$\lim_{x \to 1^-} f(x) = 3(1) - 2 = 3 - 2 = 1.$$ Thus, the left-hand limit is: $$\lim_{x \to 1^-} f(x) = 1.$$ Would you like further details on the limit calculation or how piecewise functions work? --- Here are 5 related questions: 1. How do you compute right-hand limits for piecewise functions? 2. What is the definition of a two-sided limit? 3. How do you determine if a function is continuous at a point? 4. What is the significance of approaching a limit from the left ($$x \to a^-$$) or right ($$x \to a^+$$)? 5. How can you evaluate limits that involve more complex functions, such as logarithmic or trigonometric functions? **Tip:** When evaluating limits, always check which part of a piecewise function is valid for the region you are approaching from!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Left-hand Limits
Formulas
lim x→1− (3x - 2)
Limit calculation using direct substitution
Theorems
Limit Theorem for piecewise functions
Suitable Grade Level
Grades 11-12 (Pre-calculus or early calculus)