Math Problem Statement
Let f(x) be a piecewise function defined as follows: f(x) = {-x if x <= -4, 16 - x^2 if -4 < x < 4, x - 4 if x >= 4}. Sketch the graph of this function and find the following limits if they exist: lim(x -> -4^-) f(x), lim(x -> -4^+) f(x), lim(x -> -4) f(x), lim(x -> 4^-) f(x), lim(x -> 4^+) f(x), lim(x -> 4) f(x).
Solution
Let's examine the piecewise function and find the limits requested.
The given function is defined as:
-x & \text{if } x \leq -4 \\ 16 - x^2 & \text{if } -4 < x < 4 \\ x - 4 & \text{if } x \geq 4 \end{cases}$$ We need to compute the following limits: 1. $$\lim\limits_{x \to -4^-} f(x)$$: The left-hand limit as $$x$$ approaches $$-4$$. 2. $$\lim\limits_{x \to -4^+} f(x)$$: The right-hand limit as $$x$$ approaches $$-4$$. 3. $$\lim\limits_{x \to -4} f(x)$$: The two-sided limit as $$x$$ approaches $$-4$$. 4. $$\lim\limits_{x \to 4^-} f(x)$$: The left-hand limit as $$x$$ approaches $$4$$. 5. $$\lim\limits_{x \to 4^+} f(x)$$: The right-hand limit as $$x$$ approaches $$4$$. 6. $$\lim\limits_{x \to 4} f(x)$$: The two-sided limit as $$x$$ approaches $$4$$. ### Step 1: Calculate $$\lim\limits_{x \to -4^-} f(x)$$ When $$x \leq -4$$, the function is given by $$f(x) = -x$$. So as $$x \to -4$$ from the left, we have: $$\lim\limits_{x \to -4^-} f(x) = -(-4) = 4$$ ### Step 2: Calculate $$\lim\limits_{x \to -4^+} f(x)$$ When $$-4 < x < 4$$, the function is $$f(x) = 16 - x^2$$. So as $$x \to -4$$ from the right, we have: $$\lim\limits_{x \to -4^+} f(x) = 16 - (-4)^2 = 16 - 16 = 0$$ ### Step 3: Calculate $$\lim\limits_{x \to -4} f(x)$$ Since the left-hand and right-hand limits at $$x = -4$$ do not agree ($$4 \neq 0$$), the two-sided limit does not exist: $$\lim\limits_{x \to -4} f(x) = \text{DNE}$$ ### Step 4: Calculate $$\lim\limits_{x \to 4^-} f(x)$$ When $$-4 < x < 4$$, the function is $$f(x) = 16 - x^2$$. So as $$x \to 4$$ from the left, we have: $$\lim\limits_{x \to 4^-} f(x) = 16 - (4)^2 = 16 - 16 = 0$$ ### Step 5: Calculate $$\lim\limits_{x \to 4^+} f(x)$$ When $$x \geq 4$$, the function is $$f(x) = x - 4$$. So as $$x \to 4$$ from the right, we have: $$\lim\limits_{x \to 4^+} f(x) = 4 - 4 = 0$$ ### Step 6: Calculate $$\lim\limits_{x \to 4} f(x)$$ Since both the left-hand and right-hand limits at $$x = 4$$ agree ($$0 = 0$$), the two-sided limit exists: $$\lim\limits_{x \to 4} f(x) = 0$$ ### Final Answer: 1. $$\lim\limits_{x \to -4^-} f(x) = 4$$ 2. $$\lim\limits_{x \to -4^+} f(x) = 0$$ 3. $$\lim\limits_{x \to -4} f(x) = \text{DNE}$$ 4. $$\lim\limits_{x \to 4^-} f(x) = 0$$ 5. $$\lim\limits_{x \to 4^+} f(x) = 0$$ 6. $$\lim\limits_{x \to 4} f(x) = 0$$ Would you like further details or clarification on any part of this solution? Here are five related questions to expand your understanding: 1. How do you determine if a piecewise function is continuous at a given point? 2. What is the difference between left-hand and right-hand limits? 3. How can you graphically interpret a limit that does not exist? 4. What conditions must be met for a two-sided limit to exist? 5. Can a function have a limit if it is discontinuous at that point? **Tip:** When evaluating limits of piecewise functions, always check the behavior of the function on both sides of the point in question.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
Limit of a function as x approaches a value from the left and right
Conditions for the existence of a two-sided limit
Theorems
Limit Definition
Continuity Criteria
Suitable Grade Level
Undergraduate Calculus or Advanced High School Calculus