Math Problem Statement
1. Graficar en GeoGebra la siguiente función a trozos, y de acuerdo con ella determinar los límites laterales dados, presentar la gráfica y la respuesta a cada inciso.
Graficar en GeoGebra la siguiente función a trozos, y de acuerdo con ella determinar los límites laterales dados, presentar la gráfica y la respuesta a cada inciso.
Solution
To solve this problem, let's first examine the piecewise function and understand the limits at the specified points.
Function Definition:
The function is defined as follows:
x + 2 & \text{if } x \leq -1 \\ -x^2 + 2x & \text{if } -1 < x < 1 \\ \frac{1}{x} & \text{if } x > 1 \end{cases}$$ ### Problem Requirements: We need to evaluate the following limits: 1. $$\lim_{x \to -1^-} f(x)$$ (left-hand limit as $$x \to -1$$) 2. $$\lim_{x \to -1^+} f(x)$$ (right-hand limit as $$x \to -1$$) 3. $$\lim_{x \to 1^-} f(x)$$ (left-hand limit as $$x \to 1$$) 4. $$\lim_{x \to 1^+} f(x)$$ (right-hand limit as $$x \to 1$$) I'll go through each limit analytically and then generate a GeoGebra graph for visualization. --- ### Solution: 1. **Limit as $$x \to -1^-$$:** For $$x \to -1^-$$, we use the piece $$f(x) = x + 2$$. $$\lim_{x \to -1^-} f(x) = (-1) + 2 = 1$$ 2Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Limit from the left as x approaches a point: lim(x→a^-) f(x)
Limit from the right as x approaches a point: lim(x→a^+) f(x)
Theorems
Limit laws
Continuity of piecewise functions at boundaries
Suitable Grade Level
Grades 11-12