Math Problem Statement

f(x) =

x − 6, for x ≤ −1 x2 + 1, for −1 < x ≤ 1 8 − x, for x > 1 Find each limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.) (a) lim x→−1− f(x)

(b) lim x→−1+ f(x)

(c) lim x→−1 f(x)

(d) Find f(−1).

Solution

To solve these problems, we will evaluate the piecewise function f(x)f(x) for different limits and for a specific value of xx. The function f(x)f(x) is defined as:

\begin{cases} x - 6 & \text{for } x \leq -1 \\ x^2 + 1 & \text{for } -1 < x \leq 1 \\ 8 - x & \text{for } x > 1 \end{cases}$$ Let's go through each part of the question step-by-step: ### (a) $$\lim_{x \to -1^-} f(x)$$ This represents the limit of $$f(x)$$ as $$x$$ approaches $$-1$$ from the left. For $$x \leq -1$$, the function is defined as $$f(x) = x - 6$$. So, as $$x \to -1$$ from the left: $$f(x) = x - 6 \implies f(-1) = -1 - 6 = -7$$ Therefore: $$\lim_{x \to -1^-} f(x) = -7$$ ### (b) $$\lim_{x \to -1^+} f(x)$$ This represents the limit of $$f(x)$$ as $$x$$ approaches $$-1$$ from the right. For $$-1 < x \leq 1$$, the function is defined as $$f(x) = x^2 + 1$$. So, as $$x \to -1$$ from the right: $$f(x) = x^2 + 1 \implies f(-1) = (-1)^2 + 1 = 1 + 1 = 2$$ Therefore: $$\lim_{x \to -1^+} f(x) = 2$$ ### (c) $$\lim_{x \to -1} f(x)$$ This represents the two-sided limit of $$f(x)$$ as $$x$$ approaches $$-1$$. The limit exists only if the left-hand limit and the right-hand limit are equal. From parts (a) and (b), we have: $$\lim_{x \to -1^-} f(x) = -7 \quad \text{and} \quad \lim_{x \to -1^+} f(x) = 2$$ Since these two limits are not equal, the limit does not exist. Therefore: $$\lim_{x \to -1} f(x) = \text{DNE}$$ ### (d) $$f(-1)$$ To find the value of $$f(-1)$$, we need to check which part of the piecewise function includes $$x = -1$$. For $$x = -1$$, the condition $$x \leq -1$$ applies, so we use the definition: $$f(x) = x - 6$$ Thus: $$f(-1) = -1 - 6 = -7$$ Therefore: $$f(-1) = -7$$ ### Summary of Answers (a) $$\lim_{x \to -1^-} f(x) = -7$$ (b) $$\lim_{x \to -1^+} f(x) = 2$$ (c) $$\lim_{x \to -1} f(x) = \text{DNE}$$ (d) $$f(-1) = -7$$ Would you like more details on any part, or do you have any questions? Here are some related questions you might consider: 1. What would $$\lim_{x \to 1^-} f(x)$$ be? 2. What would $$\lim_{x \to 1^+} f(x)$$ be? 3. How do you find the derivative of this piecewise function? 4. Can you graph this piecewise function? 5. What are the continuity points of $$f(x)$$? **Tip:** When evaluating limits for piecewise functions, always consider the definition of the function for values approaching from both the left and the right.

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Math Problem Analysis

Mathematical Concepts

Piecewise functions
Limits

Formulas

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Theorems

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Suitable Grade Level

Grades 11-12