Math Problem Statement
Lim x = -2^-, f(x)=?
Lim x = -2^-, f(x)=?
Solution
The given function is a piecewise function defined as follows:
x + 5 & \text{if } x < -2, \\ 5 & \text{if } x > -2. \end{cases}$$ You are asking for the limit of $$f(x)$$ as $$x \to -2^-$$, which means from the left-hand side of $$-2$$. For values of $$x$$ approaching $$-2$$ from the left (i.e., $$x < -2$$), the function follows the rule $$f(x) = x + 5$$. Thus, we compute the left-hand limit: $$\lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (x + 5) = -2 + 5 = 3.$$ Therefore, the limit as $$x \to -2^-$$ is: $$\lim_{x \to -2^-} f(x) = 3.$$ Would you like further details on this, or do you have any questions? ### Related Questions: 1. What is the right-hand limit $$\lim_{x \to -2^+} f(x)$$? 2. Is $$f(x)$$ continuous at $$x = -2$$? 3. How do we find the overall limit of a piecewise function at a given point? 4. How do limits relate to the continuity of functions? 5. Can the function $$f(x)$$ be differentiable at $$x = -2$$? ### Tip: When dealing with piecewise functions, always check the conditions for each piece and evaluate the limits from both sides to understand behavior around a given point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
f(x) = x + 5 for x < -2
f(x) = 5 for x > -2
Left-hand limit: lim_{x \to -2^-} f(x)
Theorems
Limit definition
Piecewise function continuity
Suitable Grade Level
Grades 10-12