Math Problem Statement
Let f(x) = { 10 - x - x^2 if x <= 4, 2x - 5 if x > 4. Calculate the following limits: lim (x -> 4^-) f(x), lim (x -> 4^+) f(x), lim (x -> 4) f(x). Enter 'DNE' if the limit does not exist.
Solution
We are given a piecewise function defined as follows:
\begin{cases} 10 - x - x^2 & \text{if } x \leq 4 \\ 2x - 5 & \text{if } x > 4 \end{cases}$$ We are asked to calculate three limits involving this function: 1. $$\lim_{x \to 4^-} f(x)$$ (the left-hand limit as $$x$$ approaches 4), 2. $$\lim_{x \to 4^+} f(x)$$ (the right-hand limit as $$x$$ approaches 4), 3. $$\lim_{x \to 4} f(x)$$ (the two-sided limit as $$x$$ approaches 4). ### Step 1: Left-hand limit $$\lim_{x \to 4^-} f(x)$$ For the left-hand limit, we are interested in the behavior of $$f(x)$$ as $$x$$ approaches 4 from values **less than** 4. In this region, the function is defined as $$f(x) = 10 - x - x^2$$. Substituting $$x = 4$$ into this expression: $$f(4^-) = 10 - 4 - 4^2 = 10 - 4 - 16 = -10$$ Thus, $$\lim_{x \to 4^-} f(x) = -10$$. ### Step 2: Right-hand limit $$\lim_{x \to 4^+} f(x)$$ For the right-hand limit, we consider the behavior of $$f(x)$$ as $$x$$ approaches 4 from values **greater than** 4. In this region, the function is $$f(x) = 2x - 5$$. Substituting $$x = 4$$ into this expression: $$f(4^+) = 2(4) - 5 = 8 - 5 = 3$$ Thus, $$\lim_{x \to 4^+} f(x) = 3$$. ### Step 3: Two-sided limit $$\lim_{x \to 4} f(x)$$ For the two-sided limit to exist, the left-hand limit and the right-hand limit must be equal. In this case, we found: - $$\lim_{x \to 4^-} f(x) = -10$$, - $$\lim_{x \to 4^+} f(x) = 3$$. Since these two values are not equal, the two-sided limit **does not exist**. ### Final Answers: 1. $$\lim_{x \to 4^-} f(x) = -10$$ 2. $$\lim_{x \to 4^+} f(x) = 3$$ 3. $$\lim_{x \to 4} f(x) = \text{DNE}$$ (does not exist) Would you like more details on any of these steps or concepts? --- Here are 5 related questions for further exploration: 1. How do you compute the limit of a piecewise function at a different point of discontinuity? 2. What is the significance of one-sided limits in calculus? 3. How can we prove that a function is continuous at a point? 4. How can we interpret the behavior of a function graphically when limits do not exist? 5. What are the criteria for a function to be differentiable at a point? **Tip:** When limits from the left and right don't match, the function has a **discontinuity** at that point!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
One-Sided Limits
Continuity
Formulas
Left-hand limit: lim(x -> c^-) f(x)
Right-hand limit: lim(x -> c^+) f(x)
Two-sided limit: lim(x -> c) f(x)
Theorems
For the two-sided limit to exist at x = c, the left-hand limit and right-hand limit must be equal.
Suitable Grade Level
Grades 11-12 (Advanced High School Calculus)