Math Problem Statement
Iff(x)={-4x+12x+2 , x≤5x>5 then find limx→5− f(x)
Solution
The function is given as:
\begin{cases} -4x + 12x + 2, & \text{for } x \leq 5 \\ x > 5 \end{cases}$$ You are asked to find: $$\lim_{{x \to 5^-}} f(x)$$ ### Step 1: Simplify the expression for $$f(x)$$ when $$x \leq 5$$ In the first case, the expression for $$f(x)$$ is: $$f(x) = -4x + 12x + 2$$ Simplifying this: $$f(x) = 8x + 2$$ ### Step 2: Evaluate the limit as $$x$$ approaches 5 from the left To find $$\lim_{{x \to 5^-}} f(x)$$, we need to evaluate the function $$f(x) = 8x + 2$$ as $$x$$ approaches 5 from the left (i.e., when $$x \leq 5$$): $$\lim_{{x \to 5^-}} (8x + 2) = 8(5) + 2 = 40 + 2 = 42$$ ### Final Answer: $$\lim_{{x \to 5^-}} f(x) = 42$$ Would you like further details or clarifications on any part of this solution? ### Follow-up Questions: 1. What is the value of $$\lim_{{x \to 5^+}} f(x)$$? 2. How does continuity at a point relate to left and right limits? 3. Can a function have different limits from the left and right at the same point? 4. How do you solve limits for piecewise functions? 5. How does the behavior of a function change around critical points like $$x = 5$$ here? ### Tip: When dealing with piecewise functions, always evaluate limits separately from the left and right, as the function may behave differently in different regions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Algebra
Formulas
lim_{x → c} f(x)
Piecewise function evaluation
Theorems
Left-hand limit
Continuity of functions
Suitable Grade Level
Grades 11-12