Math Problem Statement
f(x) =
−5 − x
+ 6,
if x < −6
6,
if x = −6
2x + 19,
if x > −6
Find each limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)
(a)
lim x→−6− f(x) (b)
lim x→−6+ f(x) (c)
lim x→−6 f(x)
Solution
To solve these problems, we will evaluate the piecewise function for different limits. The function is defined as:
\begin{cases} -5 - x + 6 & \text{if } x < -6 \\ 6 & \text{if } x = -6 \\ 2x + 19 & \text{if } x > -6 \end{cases}$$ First, let's simplify the function expressions: - For $$x < -6$$: $$f(x) = -5 - x + 6 = 1 - x$$ - For $$x = -6$$: $$f(x) = 6$$ - For $$x > -6$$: $$f(x) = 2x + 19$$ Let's evaluate each limit: ### (a) $$\lim_{x \to -6^-} f(x)$$ This is the limit of $$f(x)$$ as $$x$$ approaches $$-6$$ from the left. For $$x < -6$$, the function is defined as $$f(x) = 1 - x$$. So, as $$x \to -6$$ from the left: $$f(x) = 1 - x$$ Plugging in $$x = -6$$: $$f(-6) = 1 - (-6) = 1 + 6 = 7$$ Therefore: $$\lim_{x \to -6^-} f(x) = 7$$ ### (b) $$\lim_{x \to -6^+} f(x)$$ This is the limit of $$f(x)$$ as $$x$$ approaches $$-6$$ from the right. For $$x > -6$$, the function is defined as $$f(x) = 2x + 19$$. So, as $$x \to -6$$ from the right: $$f(x) = 2x + 19$$ Plugging in $$x = -6$$: $$f(-6) = 2(-6) + 19 = -12 + 19 = 7$$ Therefore: $$\lim_{x \to -6^+} f(x) = 7$$ ### (c) $$\lim_{x \to -6} f(x)$$ This is the two-sided limit of $$f(x)$$ as $$x$$ approaches $$-6$$. The two-sided limit exists if the left-hand limit and the right-hand limit are equal. From parts (a) and (b), we have: $$\lim_{x \to -6^-} f(x) = 7 \quad \text{and} \quad \lim_{x \to -6^+} f(x) = 7$$ Since these two limits are equal, the two-sided limit exists and is equal to this value. Therefore: $$\lim_{x \to -6} f(x) = 7$$ ### Summary of Answers (a) $$\lim_{x \to -6^-} f(x) = 7$$ (b) $$\lim_{x \to -6^+} f(x) = 7$$ (c) $$\lim_{x \to -6} f(x) = 7$$ Would you like more details on any part, or do you have any questions? Here are some related questions you might consider: 1. What is $$f(-6)$$? 2. What would $$\lim_{x \to 0^-} f(x)$$ be if the function had a different domain? 3. What are the continuity points of $$f(x)$$? 4. How do you graph this piecewise function? 5. Can you determine the derivatives of this piecewise function at different points? **Tip:** When evaluating limits for piecewise functions, check the function definition for values approaching from both directions to determine if the limits match.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
-
Theorems
-
Suitable Grade Level
Grades 11-12