Math Problem Statement

In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. g(x)={x3−1,1,x≤0x>0 limx→0−g(x) limx→0+g(x)

Solution

The piecewise-defined function you provided is:

\begin{cases} x^3 - 1 & \text{for } x \leq 0 \\ 1 & \text{for } x > 0 \end{cases}$$ We are tasked with evaluating the following limits: 1. $$\lim_{x \to 0^-} g(x)$$ (limit as $$x$$ approaches 0 from the left). 2. $$\lim_{x \to 0^+} g(x)$$ (limit as $$x$$ approaches 0 from the right). ### Step 1: Left-hand limit $$\lim_{x \to 0^-} g(x)$$ For $$x \leq 0$$, the function is $$g(x) = x^3 - 1$$. To find the left-hand limit, we substitute values approaching 0 from the left into $$x^3 - 1$$. $$\lim_{x \to 0^-} (x^3 - 1) = 0^3 - 1 = -1$$ Thus, $$\lim_{x \to 0^-} g(x) = -1$$ ### Step 2: Right-hand limit $$\lim_{x \to 0^+} g(x)$$ For $$x > 0$$, the function is $$g(x) = 1$$. This is a constant function for $$x > 0$$, so the limit is simply the value of the function: $$\lim_{x \to 0^+} g(x) = 1$$ ### Conclusion: - $$\lim_{x \to 0^-} g(x) = -1$$ - $$\lim_{x \to 0^+} g(x) = 1$$ Since the left-hand and right-hand limits do not match, the two-sided limit $$\lim_{x \to 0} g(x)$$ does not exist. Would you like further details or have any specific questions? Here are some related questions that might help expand on this concept: 1. How do we evaluate a two-sided limit for a piecewise function? 2. What is the significance of left-hand and right-hand limits in piecewise-defined functions? 3. How do discontinuities affect the existence of limits? 4. Can limits be used to find the continuity of a function? 5. How do the properties of polynomials influence the limit evaluation? **Tip:** Always check whether the function is continuous at the point in question. If the left-hand and right-hand limits don't match, the function is discontinuous at that point.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Limits
Continuity

Formulas

g(x) = x^3 - 1 for x <= 0
g(x) = 1 for x > 0

Theorems

Limit laws
Piecewise function limits
Left-hand and right-hand limits

Suitable Grade Level

Grades 11-12