Math Problem Statement
Find the right-hand limit of the piecewise function F(x) as x approaches 3 from the right.
Solution
The problem involves finding the right-hand limit of a piecewise function as . Here's the process to solve it:
Step 1: Analyze the piecewise function
The function is defined as:
\begin{cases} (x - 3)^2 & \text{if } x < 3, \\ e^x + 3 & \text{if } x \geq 3. \end{cases}$$ The **right-hand limit** means $$x \to 3^+$$, so we consider the part of the function where $$x \geq 3$$, which is: $$F(x) = e^x + 3.$$ ### Step 2: Compute the limit Substitute $$x = 3$$ into the expression for $$F(x)$$ as $$x \to 3^+$$: $$\lim_{x \to 3^+} F(x) = e^3 + 3.$$ ### Step 3: Write the final answer $$\lim_{x \to 3^+} F(x) = e^3 + 3.$$ ### Explanation 1. Since we are considering the right-hand limit, we use the definition of $$F(x)$$ for $$x \geq 3$$. 2. The exponential function $$e^x$$ is continuous, so substituting $$x = 3$$ directly gives the correct value. Would you like further clarifications or additional related examples? --- ### Related Questions: 1. What is the left-hand limit $$\lim_{x \to 3^-} F(x)$$, and how does it compare to the right-hand limit? 2. Is the function $$F(x)$$ continuous at $$x = 3$$? Why or why not? 3. Can you generalize the behavior of $$F(x)$$ for all $$x$$ in terms of continuity and differentiability? 4. How does the value of $$e^x + 3$$ change as $$x \to 3^+$$? Discuss its behavior for nearby values. 5. How do piecewise functions behave when their definitions change across a boundary point? --- ### Tip: When dealing with piecewise functions, always identify which part of the function applies for the given limit direction (left or right).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
lim_{x→a^+} f(x) = f(a) if f is continuous from the right
Theorems
Limit Laws
Continuity of Exponential Functions
Suitable Grade Level
Grades 11-12