Math Problem Statement
For the function f, of, xf(x) shown below, determine limit, start subscript, x, right arrow, 1, end subscript, f, of, x, . x→1 lim f(x). f, of, x, equals, left curly bracket, begin cases, begin case 1, minus, 2, x, squared, minus, 3, for , x, is less than or equal to, 1, end case 1, begin case 2, minus, 4, plus, 4, x, for , x, is greater than, 1, end case 2, end cases, . f(x)={ −2x 2 −3 −4+4x
for x≤1 1 1
for x>1 1 1
Solution
To find the limit of the function as , we need to examine the behavior of the function from both the left-hand side (as approaches 1 from values less than or equal to 1) and the right-hand side (as approaches 1 from values greater than 1).
The given piecewise function is:
\begin{cases} -2x^2 - 3 & \text{for } x \leq 1 \\ -4 + 4x & \text{for } x > 1 \end{cases}$$ We will now compute the left-hand limit and right-hand limit of the function as $$x \to 1$$. ### 1. Left-hand limit (as $$x \to 1^-$$): For $$x \leq 1$$, the function is $$f(x) = -2x^2 - 3$$. Substitute $$x = 1$$ into this expression: $$f(1) = -2(1)^2 - 3 = -2 - 3 = -5$$ Thus, the left-hand limit is: $$\lim_{x \to 1^-} f(x) = -5$$ ### 2. Right-hand limit (as $$x \to 1^+$$): For $$x > 1$$, the function is $$f(x) = -4 + 4x$$. Substitute $$x = 1$$ into this expression: $$f(1^+) = -4 + 4(1) = -4 + 4 = 0$$ Thus, the right-hand limit is: $$\lim_{x \to 1^+} f(x) = 0$$ ### 3. Conclusion: Since the left-hand limit $$(-5)$$ and the right-hand limit $$(0)$$ are not equal, the limit of $$f(x)$$ as $$x \to 1$$ **does not exist**. $$\lim_{x \to 1} f(x) \quad \text{does not exist}.$$ Would you like more details or have any questions on this problem? Here are five related questions to expand your understanding: 1. How do you determine the continuity of a piecewise function? 2. What is the importance of matching left-hand and right-hand limits? 3. How would you calculate the limit of a trigonometric piecewise function? 4. Can a function be discontinuous at a point but have a defined limit? 5. How would you graph this piecewise function? **Tip:** For piecewise functions, always check limits from both sides at the boundary points to determine if the function is continuous.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Left-hand limit: lim x→1^- f(x) = -2x^2 - 3
Right-hand limit: lim x→1^+ f(x) = -4 + 4x
Theorems
Limit Laws
Piecewise Continuity Theorem
Suitable Grade Level
Grades 10-12